DEPARTMENT OF MATHEMATICS AND STATISTICS UMASS - AMHERST BASIC EXAM - PROBABILITY January 2013 Work all problems.Show all work. Explain your answers. State the theorems used whenever possible. 60 points are needed to pass at the Masters Level and 75 to pass at the Ph.D. level. 1. Let Y1 and Y2 have the joint probability density function given by f (y1 , y2 ) = c(2 − y1 ) for 0 ≤ y2 ≤ y1 ≤ 2. For the following parts, you can leave your answers in terms of integrals with explicit limits. No need to give the final numerical answers. (a) (6 points) Find c. (b) (6 points) Find the marginal density functions for Y1 and Y2 . (c) (6 points) Are Y1 and Y2 independent? Why or why not? (d) (6 points) Find the conditional density of Y1 given Y2 = y2 and P (Y1 ≥ 1.5|Y2 = 0.9). 2. Let X ∼ N (µ, σ 2 ). (a) (6 points) Show that the moment generating function of X is MX (t) = exp(µt + σ 2 t2 /2). 2 2 2 (b) (7 points) Show that if X1 ∼ N (µ1 , σ1 ), X2 ∼ N (µ2 , σ2 ), X3 ∼ N (µ3 , σ3 ), and X1 , X2 , X3 are 2 2 2 independent, then X1 + X2 + X3 ∼ N (µ1 + µ2 + µ3 , σ1 + σ2 + σ3 ). (c) (7 points) Suppose µi and σi , i = 1, 2, 3 are known quantities. Based on X1 , X2 , X3 , construct a statistic that has a Chi squared distribution with 3 degrees of freedom, a statistics that has a t distribution with 2 degrees of freedom, and a statistic that has an F distribution with 1 and 2 degrees of freedom. P3 1 2 ¯ 2 (d) (7 points) Suppose that µi = µ and σi = σ 2 for i = 1, 2, 3. Define S 2 = 2 i=1 (Xi − X ) with 2 2 ¯ X = (X1 + X2 + X3 )/3. Show that ES = σ and ES ≤ σ . 3. Suppose (X1 , X2 , X3 ) follows a multinomial distribution with m trials and cell probabilities p1 , p2 , · · · , p3 . x1 x2 x3 ! Note that the Multinomial(m, p1 , p2 , p3 ) probability mass function is x1 !m ···x3 ! p1 p2 p3 , x1 + x2 + x3 = m, p1 + p2 + p3 = 1. (a) (7 points) Find the marginal distribution of X1 and the conditional distribution of X2 given X1 . Write down your reasoning. No mathematical proof is needed. (b) (7 points) Show that Cov (X1 , X2 ) = −mp1 p2 . X11 X12 X 1n (c) (7 points) Let X21 , X22 , · · · , X2n be iid random vectors from the above multinoX31 X32 X 3n Pn 1 ¯1, X ¯ 2 ). ¯ mial distribution. Let Xj = n i=1 Xji , j = 1, 2. Find the limiting joint distribution of (X State the theorem used. ¯ 1 /X ¯ 2 . State the theorem used. (d) (7 points) Find the limiting distribution of Yn = X 4. Suppose we have three cards. The first one is blank on both sides, the second has an X on one side and is blank on the other, and the third has an X on both sides. We run an ”experiment” where we choose one card at random and then look at one side of the chosen card at random. (a) (7 points) What is the probability that you see an X ? (b) (7 points) What is the probability that you see an X and the other side of the chosen card has an X too? (c) (7 points) Suppose we run the experiment above and we see an X . Given that outcome, what is the probability that the other side of the card has an X on it too? Page 2 UNIVERSITY OF MASSACHUSETTS Department of Mathematics and Statistics Basic Exam - Statistics Wednesday, August 29, 2012 Work all problems. 60 points are needed to pass at the Masters Level and 75 to pass at the Ph.D. level. 1. (15 points) Jane is trapped in a mine containing 3 doors. The first door leads to a tunnel that will take her to safety after 3 hours of travel. The second door leads to a tunnel that will return her to her starting point in the mine after 5 hours of travel. The third door leads to a tunnel that will return her to her starting point in the mine after 7 hours. If we assume that Jane is at all times equally likely to choose any one of the doors, what is the expected length of time until she reaches safety? 2. Suppose we are in a situation where the value of a random variable X is observed and then, based on the observed value, an attempt is made to predict the value of another random variable Y . Let g (X ) denote the predictor (i.e., if X is observed to be equal to x, then g (x) is our prediction for the value of Y ). We would like to choose g so that g (X ) tends to be close to Y . Suppose we decide to choose g to minimize E ((Y − g (X ))2 ). (That defines “best” below.) Assume that the means and variances 2 2 of X and Y , denoted by µX = E (X ), µY = E (Y ), σX = V ar(X ) and σY = V ar(Y ), Cov (X,Y ) and the correlation of X and Y , denoted by ρXY = √ 2 2 , are known. σX σY (a) (10 points) What is the best predictor of Y ? (show your calculation in detail) (b) (5 points) Consider linear predictors of Y , i.e., g (X ) = a + bX . In that case, what is the best linear predictor of Y with respect to X ? In other words, choose a and b in g (X ) = a + bX as functions of the means and variances of X and Y and the correlation of X and Y . 3. The number of defects per yard in a certain fabric, Y, is known to have a Poisson distribution with parameter λ. The probability mass function is: exp(−λ)λy P r(Y = y |λ) = , λ > 0, y = 0, 1, 2, . . . y! Suppose that λ is also an exponential random variable with mean 1 (pdf: f (λ) = exp(−λ), λ ≥ 0 and 0 otherwise.) (a) (5 points) Write down and solve an integral expression for the unconditional probability mass function for Y. (b) (5 points) Without using your answer from part a above, what is the unconditional expectation of Y ? 1 (c) (5 points) Without using your answer from part a above, what is the unconditional variance of Y ? 4. Suppose Y is a random variable with pdf g (y ), X is a random variable with pdf f (x), and U has a U (0, 1) distribution. Further, let M > 1 be a constant where f (x) < M g (x) for all x. Consider the following algorithm. (i) Sample y from g (y ) and u from a U (0, 1). (ii) If u < f (y )/(M g (y )) then ”accept” y and stop the algorithm. If not, go to (i). (a) (5 points) Show that P r(U < f (Y )/(M g (Y ))) = E (f (Y )/(M g (Y ))) where the expectation is with respect to Y. (b) (5 points) Show that the probability that the algorithm accepts y on the first try is 1/M. (c) (5 points) What is the expected number of tries the algorithm will make until it accepts for the first time? (d) (5 points) Derive the density of the accepted y. 5. Central Limit Theorem and related topics (a) (10 points) State carefully a Central Limit Theorem for a sequence of i.i.d. random variables. (b) (10 points) Suppose Xi , i = 1, . . . , 100 are i.i.d. P oisson(0.0001). What is the standard error of the sample mean? (c) (5 points) Let Y = 100 i=1 Xi . Use the Central Limit Theorem to write an expression to approximate P r(Y ≥ 1). (You do not need a number.) (e) (5 points) The P oisson(λ) moment generating function is exp(λ(exp(t) − 1)). Use that result to derive an answer part c in another way. Does that support or disagree with your answer to part d? P (d) (5 points) Is the answer to part c close to zero or close to one? Why? 2 evaluate Q00 . . then express π . . (20 points) There is more information in the joint distribution of two random variables than can be discerned by looking only at their marginal distributions. State the theorems used whenever possible. and Q11 in terms of α and β . 2. In particular. y! xk k=0 k! . y2 ) = k (1 − y2 ). otherwise.AMHERST BASIC EXAM . level 1. show that 0 ≤ Q11 ≤ π . (a) Find k. 3. for λ > 0. (20 points) Let Y1 and Y2 have the joint probability density function: f (y1 . (c) If P {X2 = 1|X1 = 0} = α and P {X2 = 0|X1 = 1} = β . e−λ λy . find the correlation between X1 and X2 in terms of α and β . (c) Are Y1 and Y2 independent? Why or why not? (d) Find the conditional density function of Y2 given Y1 = y1 . (d) In part (c).DEPARTMENT OF MATHEMATICS AND STATISTICS UMASS . (a) In general. 1. 1 . and where X2 = 1 − X1 . (20 points) Suppose that the random variable Y has a Poisson distribution with mean λ. The probability mass function is f (y |λ) = (a) Prove that ex = P∞ (b) For each case in (a).D. X1 and X2 . 2. y = 0. (e) Find P r(Y2 ≥ 3/4|Y1 = 1/2). 0 ≤ y1 ≤ y2 ≤ 1 = 0. Q00 . . Let Qab = P {X1 = a. Show your work.PROBABILITY WINTER 2012 Work all problems. (c) Suppose that Y1 and Y2 are independent Poisson random variables with means λ1 and λ2 respectively. evaluate Q11 in three cases: where X1 and X2 are independent. 60 points are needed to pass at the Masters level and 75 to pass at the Ph. (b) Find the moment generating function of Y. Explain your answers. π ). (b) Find the marginal density functions for Y1 and Y2 . each distributed binomial(1. where X2 = X1 . X2 = b}. Consider two random variables. where 0 < π < 1. 4. Prove that limn→∞ P r(|Sn /n| > 0) = 0. Let Zi = Xi − µ. . Derive P r(M > m|N > k ). i = 0.i. Assume that the Xi s are independent Bernoulli random variables with P r(Xi = 1) = π . (c) What is the the probability mass function of N ? (d) Let M = N − k. . σ )Sn converges in distribution to a standard normal distribution as n → ∞. a function of n and σ. Let the random variable Xi = 1 if the ith flip is a head and 0 otherwise. . (20 points) Suppose Xi . ii. Derive the distribution of Z = Y1 + Y2 . (e) What is the probability mass function of M ? 2 .645)? 5. (b) Find f (n. . . i = 1. + Zn . Let N be the number of flips required to get the first head (N = 1. where k > 0 is a constant integer. so that Zn = f (n. . 2. .). (20 points) Suppose we flip coins. σ ). (c) Approximately what is limn→∞ P r(|Zn | > 1. Derive the distribution of Y1 |Z = k. . (b) Use the result from part (a) and the law of iterated expectations to derive E (N ). 1? (a) Let Sn = Z1 + . . (a) What is E (N |X1 = i). n are independent and each has mean µ and variance σ 2 < ∞. 50% are known to have the disease. 3. (a) Find the conditional probability that a randomly tested individual actually has the disease given that his or her test result is positive. 60 points are needed to pass at the Masters Level and 75 to pass at the Ph. (20 PTS) A blood test is 99 percent effective in detecting a certain disease when the disease is present. level. (20 PTS) Let X1 and X2 be independent variables each having an exponential distribution with mean 1. (a) Without any calculation.D. Let Y1 = X1 /X2 and Y2 = X2 . (a) Find the moment generating function for Y .UNIVERSITY OF MASSACHUSETTS Department of Mathematics and Statistics Basic Exam . 2011 Work all problems. What is the distribution of W ? Why? (c) If a is a positive constant and b is a fixed constant. However. September 2. derive the moment generating function for W = aY . (b) If a is a positive constant. (b) Suppose instead that an individual is tested only if he or she has symptoms.5 percent of the population has the disease. (c) Find the conditional distribution of Y1 given Y2 = y2 . the test also yields a false-positive result for 2 percent of healthy patients tested.Probability Friday. derive the moment generating function of V = aY + b. (b) Find the joint probability density function of Y1 and Y2 . Find the conditional probability that a person with symptoms actually has the disease given that his or her test result is positive. who do not have the disease. Among those with symptoms. (20 PTS) Suppose that Y is uniformly distributed on the interval (0.1). Suppose 0. 1 . What is the distribution of V ? Why? 2. give the marginal distribution of Y2 . (d) Find the marginal probability density function of Y1 . 1. ii. s1 = 1. that is. Define Q = n i=1 g (X1i . (15 PTS) Consider a population consisting of 3 units with known sizes. (25 PTS) (X1 . Select the next unit with probability proportional to size from among the remaining un-sampled units. . Repeat step (ii) until a sample of size n is obtained. π2 . X2i ). (a) What is the probability that the first unit selected is the unit of size 2 (s2 = 2)? (b) What is the probability that the n=2 units selected are the units of sizes 1 and 2 (in any order)? (c) Give numerical expressions for the probabilities of sampling each possible pair of units (you need not simplify these expressions completely). for π1 . X2 ). 5. and π3 . each unit is sampled with probability proportional to its size. X2i ) be independent and identically distributed P samples with the same distribution as (X1 . (d) Give numerical expressions for the probabilities of sampling each unit in the population. πj sj Simplify the expressions in the previous part (d) as necessary to show that the PPSWOR sampling algorithm does not result in a PPS sample. Define the function g (X1 . . X2 ) where i = 1. where πi is the probability that unit i is selected (again you need not simplify these expressions completely). Select the first unit with probability proportional to size. X2 ) is a bivariate random variable and define θ = P (X1 > X2 ). X2 ) as follows: ( 1 if X1 > X2 g (X1 .4. (d) Obtain the asymptotic distribution of n−1/2 (Q − nθ). s2 = 2. Find the distribution of Q. n. X2 ))? (b) Let the pairs (X1i . from which we will select n = 2 units using a procedure known as probability proportional to size without replacement (PPSWOR) sampling. E (g (X1 . In a PPS sample. (e) This sampling scheme is sometimes used to approximate a probability proportional to size (PPS) sample. . The PPSWOR algorithm proceeds as follows: i. X2 ) = 0 otherwise (a) What is the expected value of g (X1 . s3 = 3. iii. (c) Show Q/n converges in probability to θ as n goes to infinity. 2 . . That is: πi si = . (d) Use X/N as an estimate of |A|/|B | and |B | × (X/N ) as an estimate of |A|. 60 points are needed to pass at the Masters Level and 75 to pass at the Ph. in B . uniformly.AMHERST BASIC EXAM . y ) ∈ A.PROBABILITY WINTER 2011 Work all problems.DEPARTMENT OF MATHEMATICS AND STATISTICS UMASS . If the goal is to estimate |A| as accurately as possible. 1. (c) Let X be the number of generated points that lie within A. (b) Generate N (a large integer) points at random. small. as long as it’s big enough to contain A.D. Each question is worth 20 points. The boundary of A is known and. for any (x. The following method is proposed to estimate |A|: (a) Construct a rectangle B that contains A. or somewhere in between? Justify your answer. y ) ∈ R2 it is easy to determine whether (x. Let A be a bounded region in R2 and |A| be its area. Hint: think about Binomial distributions. However. 1 . level. The user can choose B . |A| is not known and cannot be calculated analytically. what advice would you give the user for choosing the size of B ? Should |B | be large. then their numbers of arrivals are independent. waiting at the bus stop when you arrive. Let X be the number of passengers. 1/4). less than. or the same as E[T ]? (c) Find the density of T given X = 1. I. if two intervals are disjoint. Passengers.2. but you don’t know when the next is due. up to a constant of proportionality. apart from yourself. i.. arrive at the bus stop according to a Poisson process with rate λ = 2 people per hour. is E[T |X = 1] greater than. in hours. Adopt the prior distribution T ∼ Unif(0. You know that buses arrive every 15 minutes. What is the familiar density? 2 . (a) Find E[T ].e.e. Write an intuitive argument for whether that should increase or decrease your expected value for T . in any interval of length `.. (b) Suppose X = 1. other than yourself. the number of arrivals has a Poisson distribution with parameter 2` and. You go to the bus stop to catch a bus. since the previous bus. Let T be the time elapsed. It is a truncated version of a familiar density. X1 = x1 ) = Pr(X3 = x3 |X1 = x1 ). . X2 = x2 . constant for all n ≥ 0. {X0 . X2 . . . whenever possible. X1 = x1 . For full credit. . A discrete-time Markov chain is a series of indexed random variables. (a) Give an expression in terms of p for the probability that Xn+2 = j given Xn = i. Pr(X3 = x3 |X0 = x0 . X1 . namely Pr(Xn+1 = j |X0 = x0 . .3. (c) Prove that for any Markov chain. Consider such a Markov chain in which there are only finitely many possible x’s and in which the so-called transition probabilities are given by the matrix p such that pij = Pr(Xn+1 = j |Xn = i). . 3 . express your answer using matrix notation rather than functions of the matrix elements.} which displays the Markov property. Xn = i) = Pr(Xn+1 = j |Xn = i). (b) Give an expression in terms of p for the probability that Xn+m = j given Xn = i. x2 ) = 0 otherwise. (a) Draw a picture to show the x1 and x2 values where the density is non-zero. purely graphical solutions will not get full credit. Except for part (a).4. Suppose X1 and X2 are random variables with joint density function f (x1 . x2 ) = c when x1 + x2 ≤ 1 and both x1 and x2 are non-negative. The density f (x1 . (b) What is c? (c) What is the probability that X1 > X2 ? (d) Are X1 and X2 independent? Why or why not? (e) What is the density of Y = 1/X1 ? 4 . Find the E (X1 |X1 > c). x > c and 0 otherwise. Show that the density of X1 |X1 > c is exp(−x)/ {1 − exp(−c)} . (c) Suppose λ = 1. Find the moment generating function of X1 . and f (x) = 0 otherwise.5. Let c > 0. Suppose X1 and X2 are independent and identically distributed random variables with density f (x) = λ exp(−λx). (d) Suppose λ = 1. y ≥ 0. (a) The moment generating function of a random variable X is MX (t) = E[etX ]. x ≥ 0. (b) Use the moment generating function to show that Y = X1 + X2 has density f (y ) = λ2 y exp(−λy ). 5 . and f (y ) = 0 otherwise. Let c > 0. and X2 has an exponential distribution with mean 2.f. i = 1. . level. σ 2 ). · · · . (c) Let X1 and X2 be two independent exponentially distributed random variables. Uk ) are independent. ¯ and S 2 Hence. . · · · .D.AMHERST BASIC EXAM . 60 points are needed to pass at the Masters Level and 75 to pass at the Ph. · · ·. k (a) Show that U = (U1 . Y3 be the number of additional tosses required to get a face different than the first two distinct faces. Suppose that X1 . 1. Let Z be a standard normal random variable and let Y1 = Z and Y2 = Z 2 . 6. (a) Let Y1 be the trial on which the first face is tossed. Find P (X1 > X2 |X1 < 2X2 ). and E (Y1 Y2 ). Each question is worth 20 points. Y2 be the number of additional tosses required to get a face different than the first. · · · .DEPARTMENT OF MATHEMATICS AND STATISTICS UMASS . (b) What is the expected number of tosses required in order to observe each of the six faces? 2. Uk alone. Are Y1 and Y2 independent? 4. 3. k ≥ 2. . (b) Find Cov (Y1 .f. (Hint: You may use the fact that S2 = 2 1 2 1≤i<j ≤k 2 (Xi − Xj ) ).PROBABILITY FALL 2010 Work all problems. (a) Find E (Y1 ). (b) Show that U1 and (U2 . show that X k are independently distributed. E (Y2 ). · · · . 1 . of Y = X 2 . Find the distribution of each Yi. Find the p. 1). Uk ) has a k -dimensional normal distribution. Suppose you are told to toss a die until you have observed each of the six faces. · · · . Uj = X1 − Xj for j = 2.d. Y2). (b) Let X1 and X2 be two independent random variables. X1 has an exponential distribution with mean 1. each with mean 1. Xk are iid N (µ.. (c) Express S 2 as a function of U2 . and Y6 be the number of additional tosses required to get the last remaining face after all other faces have been observed. of Y = 2X1 + X2 . Denote: k U1 = i=1 Xi . (a) Suppose X ∼ N (0. Find the p.d. (Hint: you may use the theorem that says if Xn converges to X in probability and Yn converges to Y in probability.5. If further X = a and Y = b are constants. (b) Let Xn = n with probability 1/n 0 with probability 1 − 1/n. Y ) in probability. then f only needs to be continuous at (a.). and if f is continuous. then f (Xn . (a) Let {ξn . Prove that ξ1 + ξ2 + · · · + ξn µ → 2 . but E (Xn ) and V ar (Xn ) do not converge to zero. 2 . b). V ar (ξ1 ) = σ 2 < ∞. 2 2 2 ξ1 + ξ2 + · · · + ξn µ + σ2 n → ∞. in probability. n ≥ 1} be a sequence of independently identically distributed random variables with E (ξ1 ) = µ. Show that Xn converges in probability to zero. Yn ) converges to f (X. and P (ξ1 = 0) = 0. each parent’s gene has probability 0. and aa. Thus there are three possible genotypes: AA. according to the genotype of the father.5 of being chosen. A gene has two possible forms (alleles): A and a. for some function A(θ). respectively. Assume that F is independent of M . Suppose that X has such a density. respectively (q = 1 − p). Hint: Ry you may use the fact that y = 0 dz . M . and q 2 . Show E[Y ] = 0 Pr[Y ≥ y ] dy . P∞ (b) A discrete random variable X takes values on the positive integers 1. ∞).e. 3. (a) Show that the moment generating function MX | θ (t) = E[etX ] is e[A(θ+t)−A(θ)] . Compute the nine probabilities pik in terms of p and q . Let Y ~ and covariance matrix Σ. 1 .) Let pik = Pr[C = k | M = i]. let the random variables F . Let Y be the area of a circle of radius X . Suppose that X is a random variable with density (3x + 1)/8 on the interval (0. 2. Y3 2 0 ρ 1 (b) What is the distribution of Y1 + Y2 + Y3 . 2). the mother’s contribution is independent of the father’s contribution. a mother. A family of densities is called a univariate natural exponential family if. For example. . . . . 2. ~ have a trivariate Gaussian distribution with mean vector µ 2. (a) For which values of ρ are Y1 + Y2 + Y3 and Y1 − Y2 − Y3 statistically independent? 5. state theorems used whenever possible. and C denote the genotypes of the father. Assume that their proportions in the population are p2 . Find the density of Y . the conditional probability that the child is of type k given that the mother (or father) is of type i. that the population mates randomly. For a family with a father. 2pq . aA. Σ = ρ 1 ρ . and that the conditional distribution of C given (F. F is either 1. 1. and 3. or 3. (b) Show that E[X ] = A0 (θ).Department of Mathematics and Statistics Basic Probability Exam January 2010 Work all problems. Show E [X ] = x=1 Pr[X ≥ x]. i. respectively. (Children inherit one gene from each parent. mother. including its name and associated parameters. µ Y ~ = −1 . Show your work. the density of X given θ can be expressed as p(x | θ) = h(x)eθx−A(θ) . Number them 1. explain your answers. M ) is determined by the familiar rules of genetics. (a) A continuous random variable Y takes values on the interval (0. R∞ 4. and child. where Y1 1 1 ρ 0 ~ = Y2 . and one child. 2. E(Z) (c) (15 pt) Find the variance of Z . . . the test also yields a false-positive result for 2 percent of the healthy patients tested.e. What is the standard ¯ . (a) (5 pt) Find the conditional distribution of Z given U = u (b) (5 pt) Find the expected value of Z . Unif(0.d.. Find the conditional probability that a randomly tested individual actually has the disease given that his or her test result is positive. depending on X such that Y = (−2)X . State the theorems used whenever possible. five red and four white. (a) (6 pt) Find the distribution of X .56. (b) (5 pt) Suppose X1 . (15 pt) A blood test is 99 percent effective in detecting a certain disease when the disease is present. 1). (10 pt) What is the pdf of Y 2 ? −∞<y <∞ 3. Explain your answers.1). Three are drawn out at random without replacement. (5 pt) Find the expected value and variance of Y . let X denote the number of red chips in the sample. . Let X1 and X2 be independent standard normal random variables. X100 ? deviation of X (c) (15 pt) Use the Central Limit Theorem to find approximately the probability that the average of the 100 √ numbers chosen exceeds 0. 2. (b) (4 pt) Compute the expected payment of a player. Consider a random variable Y with probability density function (pdf) given by f (y ) = ce−y (a) (b) (c) (d) 2 /2 . An urn contains nine chips. i. X100 ∼ i. random variables. Suppose 0. . and assume that U is uniformly distributed over (0. 60 points are needed to pass at the Masters level and 75 to pass at the Ph. that of Y .D.5 percent of the population has the disease. (5 pt) Find c. Define Z = U X1 + (1 − U )X2 . who have no such disease. However. V(Z) 4.d. level 1. Let U be independent of X1 and X2 . (5 pt) Derive the moment-generating-function of Y . . the mean of X1 . 1 . .Department of Mathematics and Statistics Basic Probability Exam August 2009 Work all problems. You may use the approximation 1/ 12 ≈ 0.i. (a) (5 pt) State carefully the Central Limit Theorem for a sequence of i.3. 5. .i. Let Y denote the payment in dollars received by player. . Show your work. Other quadrats have good salamander habitat. 0. During the study.4.) In a type A forest.D. Explain your answers. In those quadrats the number of salamanders is 0. There are two types of forest. and 0. Type A is conducive to salamanders while type B is not. In a type B forest the probability that a quadrat is good is 0. level 1. Types A and B are equally likely. (Yes. It has 0 salamanders. what is the probability that the forest is type A? (e) 4 pts Now the ecologists prepare to sample the second quadrat. they randomly sample quadrats.7.2. 1 salamander. Show your work. 1. what is the probability that a quadrat has 0 salamanders. (A quadrat is a squaremeter plot. What is the probability that the quadrat is good? (d) 4 pts Given that the quadrat had 0 salamanders. (a) 4 pts On average. with probabilities 0. what is the probability that the second quadrat is good? (f) 4 pts Given the results from the first quadrat. 0. 60 points are needed to pass at the Masters level and 75 to pass at the Ph.3. there might be no salamanders in a quadrat with good habitat.3 and the probability that it is poor is 0. 2 salamanders. In those quadrats the number of salamanders is either 0. the probability that a quadrat is good is 0. what is the probability that they find no salamanders in the second quadrat? 1 .) In each quadrat they count the number of salamanders. State the theorems used whenever possible.Department of Mathematics and Statistics Basic Probability Exam January 2009 Work all problems.8 and the probability that it is poor is 0. what is the probability that a quadrat is good? (b) 5 pts On average. Given the results from the first quadrat. respectively. 3 salamanders? (c) 4 pts The ecologists sample the first quadrat. They are studying one forest but don’t know which type it is. Some quadrats have poor salamander habitat. or 3. 2.2.1. Ecologists are studying salamanders in a forest. x! (a) 11 pts Let X be Poisson with mean µ. k=1 (c) 7 pts What is the distribution of Y ? 2 . find the moment generating function of Y = n X Xk . µn .2. Xn are independent Poisson variables with means µ1 . Compute the moment generating function of X . . . . (b) 7 pts If X1 .d. . . A Poisson random variable with mean µ has the following p. . .f: f (x) = e−µ µx . . . for x = 0. It is known that: ey = ∞ X yk k=0 k! . 1. 2. . . (a) 12 pts Show that the moment generating function of X is: MX (t) = exp(µt + σ 2 t2 /2) 2 2 (b) 8 pts Show that if X1 ∼ N(µ1 . then X1 + X2 ∼ N(µ1 + µ2 .3. σ2 ) and X1 and 2 2 X2 are independent. σ1 ) and X2 ∼ N(µ2 . 3 . σ 2 ). Let X ∼ N(µ. σ1 + σ2 ). −1). and (1. 1). X 2 ) be distributed uniformly on the disk where X1 + X2 ≤ 1. 1). i. i. (−1. −1). iv. (a) Let (X1 . 15 pts Find the joint density p(r. x2 )? 4 pts Are X1 and X2 independent? Explain. iii. (b) Let (X1 . Let R = X1 2 Θ = arctan(X1 /X2 ). (−1. Hint: it may help to draw a picture. p 2 2 Let R = X1 + X2 and Θ = arctan(X1 /X2 ). 1 pt What is the joint density p(x1 . x2 )? ii. X2 ) be distributed uniformly on the square pwhose corners 2 + X 2 and are (1. 4 pts Are R and Θ independent? Explain. 1 pt Are X1 and X2 independent? Explain. ii. iii. 4 . 4 pts Are R and Θ independent? Explain. θ).2 2 4. 1 pt What is the joint density p(x1 . given that Y1 = y1 . (a) (6pts) Find the moment generating function (MGF) of X . 3. Let Y1 = X1 + X2 and Y2 = X1 − X2 . (d) (7pts) Assuming that the number of phone calls to the switchboard is independent from hour to hour. You do not need to evaluate the integral. (c) (7pts) Use the preceding part to calculate E (X ). received by a switchboard at a specific company is Poisson with parameter λ. (a) (6pts) Find the joint pdf of Y1 and Y2 . identically distributed random variables that have the common probability density function: f (x) = exp(−x).D.95. State the theorems used whenever possible.PROBABILITY August 2008 Work all problems. X100 be independent random variables. The hourly number of phone calls. X . how many hours would be expected to have exactly 3 phone calls during a 24-hour period? 1 . Find δ 100 ¯ | < δ ) ≈ 0. λ varies independently from hour to hour according to the following exponential distribution: f (λ) = θ exp(−θλ). Answers to the following questions may depend on θ. (b) (6pts) Compute the mean and variance of X using the MGF obtained in (a). (c) (6pts) Let X1 . Show all work.AMHERST BASIC EXAM . level. (b) (6pts) Find the marginal pdf of Y1 . (c) (6pts) Find the conditional pdf of Y2 . Explain your answers. · · · .DEPARTMENT OF MATHEMATICS AND STATISTICS UMASS . (b) (7pts) Find E (X |λ) and its distribution. (a) (7pts) Find an integral expression for P (X = 4|λ ≤ 6). such that P (|X (d) (6pts) Find the distribution of Y = |X |. 60 points are needed to pass at the Masters Level and 75 to pass at the Ph. x > 0. Let X have the double exponential distribution with density f (x) = 1 2 for −∞ < x < ∞. 2. all of which 100 ¯ = 1 have the double exponential distribution. λ > 0. exp(−|x|) 1. Let X1 and X2 be independent. Let X i=1 Xi . 4. 0 < x < 1. ¯ n − 1/(1 + β )) converges ¯ n = n−1 n Xi . Show that √n(X (c) (7pts) Let X i=1 in distribution and say what it converges to. Xn be iid with density f (x) = β (1 − x)β −1 . (a) (7pts) Find P (nν (1 − X(n) ) < x) for any fixed ν and x ∈ (0. Find an approximate distribution of Tn as n goes to infinity. (b) (9pts) State the definition of convergence in distribution and find a value of ν so that nν (1 − X(n) ) converges in distribution. 2 . Define X(n) = max1≤i≤n Xi . 2 ¯n (d) (7pts) Let Tn = X . · · · . Let X1 . 1). (e) (5 pts) What is the PDF of 3Y ? 1 .AMHERST BASIC EXAM . Let G−1 (p) be the inverse of the CDF. 3. What is the probability that the individual is diseased? 2. 1. 1).001. and the test is positive. 60 points are needed to pass at the Masters Level and 75 to pass at the Ph. Suppose X ∼ U (0. What is the probability that she is healthy? (b) (5 pts) Given that an individual is diseased. 3. level. (a) (5 pts) What is c? (b) (5 pts) Derive the moment generating function of Y. √ (a) (5 pts) What is the probability that X is less than 1/2? (b) (10 pts) What is the second moment of 1/X ? (c) (10 pts) Suppose that G(y ) is the cumulative distribution function (CDF) of a continuous random variableRwith probability distribution y function (PDF) g (y ). Note that f (x) = 1 when 0 ≤ x ≤ 1 and f (x) = 0 otherwise.099. Let Z = G−1 (X ).009. 2. Note that G(y ) = −∞ g (t)dt.D.is 0. what the probability that the test is +? (c) (5 pts) Suppose an individual gets the test.DEPARTMENT OF MATHEMATICS AND STATISTICS UMASS . Suppose Y has PDF f (y ) = c exp (−y/2) when y > 0 and f (y ) = 0 otherwise. (a) (10 pts) Suppose an individual is randomly selected from that particular population. • and the probability an individual is H and test is .PROBABILITY WINTER 2008 Work all problems. A medical diagnostic test can say that an individual is diseased (+) or healthy (-).891. • the probability an individual is H and the test is + is 0. Suppose an individual can be either diseased (D) or healthy (H). Derive the PDF of Z.is 0. • the probability an individual is D and the test is + is 0. It is OK to cite a theorem. (d) (5 pts) Prove that E (Y k ) ≥ E (Y )k for k > 1. (c) (5 pts) Use the moment generating function to find E (Y k ) for k = 1. In a particular population. • the probability an individual is D and the test is . you may just state a theorem. 4. 2 . . Pn (a) (5 pts) Let Mn = n−1 i=1 Zi .(f) (5 pts) Let X = Y 1Y >3 − 3 where 1Y >3 = 1 if Y > 3 and 0 otherwise. Prove that limn→∞ P r(|Mn | > 0) = 0. Derive E (X ). are independent and identically distributed with mean µ and variance σ 2 < ∞. . i = 1. State and apply a theorem that will allow you to determine limn→∞ P r(f (n)Mn < A). Find an f (n) so that the variance of f (n)Mn = 1. . For partial credit. Suppose Xi . (b) (10 pts) Let f (n) be a function of n. Let Zi = (Xi − µ)/σ. (c) (10 pts) Let A be a constant. 3. . Suppose X ∼ Bin(n. 1).i.PROBABILITY FALL 2007 Work all problems. y ) = 1. 10). 60 points are needed to pass at the Masters Level and 75 to pass at the Ph. . 1. (b) (5 pts) What is P r(X > Y )? (c) (5 pts) What is P r(X/Y = 1)? 4. (a) (15 pts) Find the conditional distribution of X given that X + Y = j. i = 1. . . and zero otherwise. ind. p). (d) (5 pts) Find E(Y |X = x). Find an expression that involves X and known constants that converges to a N(0. x = 0.Y (x. 2.DEPARTMENT OF MATHEMATICS AND STATISTICS UMASS . i.5) ind. 0 ≤ x ≤ 10.d. y ) = 0 otherwise.Y (x. . level.) (a) (5 pts) What is the mean of X ? (b) (10 pts) What is the marginal distribution of X ? (c) (10 pts) What are E(XY ) and Var(XY )? 1 . E (X ) = 5 and V ar(X ) = 100/12. . y ) is a density.D. Give the probability mass function of this conditional distribution and identify it by its family name and parameters. . (e) (5 pts) Find Pr(X + Y < 0.AMHERST BASIC EXAM . What theorem is your result based on? (c) (10 pts) Find the mean of 1/Xi . x ≤ y ≤ x + 1. 0 ≤ x ≤ 1. . Suppose Xi ∼ U (0. . Let X and Y be random variables with pdf: fX. n. Note that the Bin(k. 0 ≤ q ≤ 1. p) and Y ∼ Bin(m. . 2. k. and fX. Pn (b) (10 pts) Let X = n−1 i=1 Xi . (a) (5 pts) Show that f (x. (a) (5 pts) Write down an expression for the probability that all Xi s are greater than 1. (The P oisson(λ) pmf is f (x) = exp(−λ)λx /x! when λ > 0 and x = 0.1) distribution as n gets large. Suppose X |Y = y ∼ Poisson(y ) and Y ∼ Unif(0. (b) (5 pts) Are X and Y independent? Why or why not? (c) (5 pts) Find fX (x). Note that f (x) = 1/10. . q ) probk x ability mass function is x q (1 − q )k−x . 1. . less than. define a function of Y1 . Find a µ and σ 1/ 2 b so that n (αn − µ)/σ converges in distribution to a standard normal. What is the moment generating function of X ? ii. (20 points) Let X be a random variable with E (X ) = µ and Pr(X = µ) < 1. Yn be a random sample from a Poisson distribution with rate λ P and P r(Yi = k ) = exp(−λ)λk /k !. or greater than exp(EX ) = exp(µ) in general? Why? (b) Now. (c) Given that Y n converges to λ in probability as n goes to infinity. .) (b) Derive the first and second moments of X. x ≥ 0. We run an ”experiment” where we choose one card at random and then look at one side of the chosen card at random. Use the previous result (part i) to derive E {exp(X )} in this case. (Note that there is more than one right answer. . b n = n −1 n (b) Let Zi = 1 if Yi > 0 and Zi = 0 otherwise. . suppose X ∼ N (0. (a) What is the probability that you see an X? (b) What is the probability that you see an X and the other side has an X too? (c) Suppose we run the experiment above and we see an X. (25 points) Let Y1 . what is the probability that the other side of the card has an X on it too? 3. . . (15 points) Let X have an exponential distribution with CDF 1 − exp(−x/λ). and the third has an X on both sides.DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS BASIC EXAM – PROBABILITY January. Let α i=1 Zi . and then use it to argue that n (Y n − λ)/ λ converges in distribution to a standard normal. Yn that converges in probability to σ in the previous question as n goes to infinity. 1) with density √1 2π i. 2007 Work all problems. P 1/2 (a) State √ the central limit theorem in general. Let Y n = n−1 n i=1 Yi . 5. (25 points) Suppose that X has density function f (x) = b(1 − x2 ). exp(−x2 /2). level. The first one is blank on both sides. λ > 0. 4. and name the results that you use to show convergence in probability. . . 1. |x| < c.D. 2. the second has an X on one side and is blank on the other. (a) Is E {exp(X )} equal to. 1. 60 points are needed to pass at the Master’s level and 75 to pass at the Ph. k = 0. Explain your answer. . . (15 points) Suppose we have three cards. Given that outcome. (a) Find constants b and c so that f (x) is a density function. (c) Derive the conditional density of X given that X is greater than 0. . . and f (x) is zero otherwise. . Show that the 1 distribution of Z = X + Y is gamma(2. λ) with density f (z ) = Γ(2) z exp(−z/λ).(a) What is the pdf of X ? (b) Suppose that Y is independent of X and has the same distribution. λ2 . Let Y1 . Let θ i=1 (b) (15 pts) Define a τ bn that converges in probability to τ as n goes to infinity. . (b) (10 pts) Find V ar(Y ). x 1 (b) (15 pts) Suppose X ∼ Exp(λ) with pdf f (x. Yn be a random sample from some distribution with P r(Yi = bn = n−1 Pn Yi .5 = (EX )0. Explain your answer. . β ) = xα−1 exp(−x/β ) . Let P r(Y = 1|X = 1) = r and P r(Y = 1|X = 0) = s. (a) (10 pts) Find P r(Y = 1) and E (Y ). What is (b) (10 pts) Find a distribution for X so that P = f (X ) has a Unif(0. (a) (15 pts) State the central limit theorem in general.DEPARTMENT OF MATHEMATICS AND STATISTICS UMASS . (a) (10 pts) Does E X 0.AMHERST BASIC EXAM . give an inequality.5 0. 1. Γ(α)β α 1 .x > 1 . you should prove that your choice works too. . Define τ as a function of θ. What is E X ? Hint: It may help to recall that the gamma distribution is g (x.PROBABILITY FALL 2006 Work all problems. level. (c) (5 pts) What is the distribution of Z = X/(Y + 1)? 4.5 = λ0. λ) = λ exp − λ . . α.D. 2. −∞ < x < ∞. Let X be a random variable with E (X ) = λ and P r(X = λ) < 1. For full credit. and then use it bn − θ)/τ converges in distribution to a standard to argue that n1/2 (θ normal. Let Y be another random variable that can also be either zero or one. The logistic function is p = f (x) = 1 1+exp(−x) . λ > 0. Let X = 1 with probability p and X = 0 with probability 1 − p. 60 points are needed to pass at the Master’s level and 75 to pass at the Ph. 3. σ 2 ) with pdf the distribution of P = f (X )? exp ⇣ x2 −2σ 2 ⌘ . and name the results that you use to show convergence in probability.1) distribution.5 ? Why or why not? If not. 0 < θ < 1. 0 < p < 1. 1 √ σ 2π (a) (10 pts) Suppose X ∼ N (0. 1) = θ. ) . (a) Over the course of the season of 100 trips. and the probabilities that either player wins a trial are: P r(A wins a trial) = p P r(B wins a trial) = q = 1 − p. or equal to log(49) or do you need more information? (and why?) (c) Next. the mean number of fish caught on any particular trip is 49. Is the mean of Yt greater than. What is the probability that no fish are caught on at least one out of 250 trips? (You may leave your answer as a formula.1) (x)I(0. Due to market forces suppose that Yt = log(Xt ). what is the approximate probability that the mean number of fish that are actually caught is no less than 48? (You may leave your answer as a formula. (25 points) Suppose that X and Y have joint density function f (x. (15 points) Let X1 and X2 be iid with Uniform(-1. 60 points are needed to pass at the Master’s level and 75 to pass at the Ph. Find the unconditional mean and variance of T . (a) What is the probability that A wins in exactly 6 trials? (b) What is the probability that A wins in exactly 2n trials? (c) What is the probability that A wins? 5. Suppose that the number of fish caught is independent from trip to trip. and the variance is also 49. make the additional assumption that the number of fish caught on a particular trip has a Poisson distribution.DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS BASIC EXAM – PROBABILITY August 31. y ) to be a density function? (b) What are the mean and variance of X ? (c) Find the conditional distribution of Y given X = x. (20 points) Two players. A and B . 2005 Work all problems. 2. level. and the first player to win two more trials than the other will win the game. (25 points) Let Xt be the number of fish that a particular fishing boat catches on trip t. The game consists of a series of trials. Suppose that each trial is iid. 3. (b) Let Z = X1 X2 . Are Y and Z independent? Why or why not? 4. y ) = c(x+y )I(0.) (b) Let Yt be the profit from trip t. (a) What must the constant c be in order for f (x.1) (y ).1) distributions. less than.D. (15 points) Suppose the number of customers Y entering a bank in a one hour period is distributed Poisson with mean 10 (pmf: e−10 10y /y !). 1. 2 (a) Find the pdf of Y = X1 . are playing a game. Suppose that given Y = y the total time T needed to service the y customers has an exponential distribution with mean 3y (pdf: e−t/(3y) /(3y )). On sunny days the number caught are iid with mean and standard deviation µs and σs . What are the marginal mean and variance of the number of fish caught under this model? . Suppose that the probability of an overcast day is p.(d) A slightly more sophisticated model posits that on overcast days the expected number of fish caught are iid with mean µo and standard deviation σo . (a) What is the probability that the weed survives? (b) What is the distribution of Z given that the weed is not killed? (c) Derive the moment generating function for Z given that Y = 1. Note that Z is random and X is fixed. Suppose the weed has an unobserved natural tolerance to the weed killer (denoted by Z ). . cdf (Φ(·)). X2 be independent Poisson variables with means µ1 . Further. and assume that this tolerance has a standard normal distribution. (b) Let X1 . . The weed either survives (Y = 1) or dies (Y = 0). Sixty points are needed to pass at the Master’s level and seventy-five at the Ph. E (Z |Y = 1) = 1 φ(−X ) φ(X ) = . x! (a) Let X be Poisson with mean µ.DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS BASIC EXAM: PROBABILITY JANUARY 2005 Work all problems. and let a1 . y ) = c.D. Compute the moment generating function of X. It may help to remember that: exp(y ) = ∞ X yk k=0 k! . You may express your answer as an unsimplified integral that involves the standard normal pdf (φ(·)). µ2 . What is the moment generating funcP2 tion of Y = i=1 ai Xi ? (c) What is the distribution of Y ? 2. 3. (b) What is the marginal pdf of X ? (c) Are X and Y independant? Why or why not. (a) Find c. x = 0. . (20 points) A weed is exposed to a known dose of weed killer (X ). 1 − Φ(−X ) Φ( X ) (d) Use the result from the previous part to derive: . level 1. 1. 0 ≤ x ≤ y ≤ 1. (20 points) Let X and Y have the joint density function f (x. (20 points) A Poisson random variable with mean µ has pmf: f (x) = exp(−µ)µx . a2 be positive constants. suppose that the weed survives if an only if Z > −X. and other functions. 2. (20 points) A game is played with n coins. flipped. (a) What are the mean and variance of Di ? 2 . suppose Ti and Fi are mutually independent. Fi . Further.4. What is the probability that it is the unfair coin (the nth coin)? 5. Write an approximation for the probability that D100 is negative. The nth coin has two heads. and replacing them back into the bag. flipping them. The commute from work. P100 (b) Let D100 be the mean difference over 100 days: D100 = i=1 Di /100. has mean µT and variance σT 2. it always lands heads up. Ti . (20 points) Joe walks to and from work each day. Let Di = Ti − Fi . The game consists of drawing coins blindly from the bag. The commute to work. has mean µF and variance σF 2. (a) Let T be the number of coins that must be drawn and flipped until one sees a total of 3 tails. and it lands heads. What is the mean of T ? (b) What is the probability that T strictly exceeds 6? (c) Suppose one coin is drawn from the bag. Coins 1 through n − 1 are “fair” and land heads with probability 1/2. . . 2 and 3 and large inventory has been built up which consists of 30% from 1. . Sixty points are needed to pass at the Master’s level and seventy-five at the Ph. 20% from 2 and 50% from 3. 3. . (b) Suppose that each item has a lifetime. . Let T denote the total lifetime of the 100 items selected (see part a). . Suppose 100 items are selected. parameterized by η = (η1 . 2 or 3. ηd ) in an open interval in Rd .2 and . With a large inventory we will treat the selections as independent where on each draw the probability is . (c) Derive the mean and variance of X . Find the expected value and variance of T . X2 . . (20 points) Let X have a Poisson distribution. is given by gη (x) = h(x) exp{ where it is assumed that Z Rd d X i=1 ηi Ti (x) − K (η )}. i = 1. h(x) exp{ d X i=1 ηi Ti (x)}dx < +∞. X2 .PROBABILITY FRIDAY. (20 points) Let X = (X1 . (20 points) Suppose a plant is manufacturing a product using three different machines 1. SEPTEMBER 3.8 for machine 3. (a) Give the probability mass function for X . 2. . Xn ) be a vector valued random variable and let Ti = Ti (X ).DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS BASIC EXAM . level. mean 6 and standard deviation . 1. and the distribution of lifetimes has mean 5 and standard deviation 1 for machine 1.3.5 of getting an item from machine 1. (a) Derive the joint distribution of X1 . (You can do this using b) if you want. and mean 7 and standard deviation . (b) Derive the moment generating function of X . d X i=1 (a) Show that K (η ) = log (b) Show that ⇣Z h(x) exp{ ηi Ti (x)}dx ⌘ E [Ti ] = 1 ∂ K (η ). .D. Suppose that the probability density function of X .5 for machine 2. . X3 where Xj = the number of itemsselected from machine j . but you don’t have to. . ∂ηi . respectively. d.) 2. 2004 Work all problems. ∂ηi ∂ηj 4. (a) Derive the joint distribution (giving the joint density suffices) of (Y1 . E [Yi ] = µY . . random sequences with E [Xi ] = µX . Y Identify and justify the limiting distribution of √ ¯ − µX ) + Y ¯ n(X as (n → +∞).i. We denote ¯ = n−1 (X1 + . . 2 .d. . Var(Xi ) = σX 2 > 0. . Define Y1 = X1 + X2 and Y2 = X1 . (15 points) Let (X1 . . (c) Give an approximation to the variance of Y1 /Y2 . . Var(Yi ) = σY 2 > 0. (25 points) Let X1 and X2 be independent exponential distributions with mean 1. + Yn ). . . Yn ) be two different i. Tj ) = ∂2 K (η ). Y2 ) (b) Find the conditional distribution of Y1 given Y2 = y2 . Xn ) and (Y1 . X ¯ = n−1 (Y1 + . . 5. + Xn ). .(c) Show that cov(Ti . . You can appeal to well known results but state clearly what results you are using and how they apply here. how many days would be expected to have exactly 2 visits over the course of a year? 5. 4. (a) Show the MGF of X is MX (t) = eµt+σ 2 t2 /2 . level. is Poisson with parameter Λ. Let Y1 = X1 /X2 Y2 = X2 (a) Without any calculation give the marginal distribution of Y2 and the conditional distribution of Y1 given Y2 = y2 .D. 2 2 (b) Show that if X1 ∼ N (µ1 . X .∞) (λ)ce−cλ (b) What is the distribution of the random variable E (X |Λ) (a) Find an integral expression for P (X = 3. (20 points) Suppose each of 100 genes has probability 0. σ 2 ). (c) Find the marginal pdf of Y1 . (20 pts) Let X ∼ N (µ. σ1 ) and X2 ∼ N (µ2 . Don’t evaluate the integral. and the genes act independently. (20 pts) The daily number of visits to a particular website.2 of mutating in a given time period. (b) State the CLT (central limit theorem) and use it to approximate P (N < 10).DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS BASIC EXAM – PROBABILITY August 27. 1. (a) What is the probability that the subsequence of boys is ascending in height and the subsequence of girls is ascending in height. (20 pts) Let X1 and X2 be independent random variables each having an exponential distribution with mean 1. σ2 ) with X1 and X2 independent then 2 2 X1 + X2 ∼ N (µ1 + µ2 . 60 points are needed to pass at the Master’s level and 75 to pass at the Ph. Λ ≤ 5). 2003 Work all problems. σ1 + σ2 ) 3. (a) Derive the exact distribution of N . Let N = the number of the genes which mutate. (b) Find the joint pdf of Y1 and Y2 . . (b) What is the probability that the sequence alternates between boys and girls? (c) What is the probability that the three boys are not together? 2. Λ varies independently from day to day according to an exponential distribution: f (λ) = 1(0. (c) Use the preceding part to compute E (X ). (d) Assuming the number of visits to the website is independent from day to day. (20 pts) Suppose 3 boys and 3 girls stand in a line in random order. Note that X is a fixed quantity. of Px and Py ? (b) Create the point Q by rotating P clockwise by θ radians around the origin. the number of those surviving. In particular. (a) What is the joint density function.DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS BASIC EXAM – PROBABILITY August 28. (20 pts) Choose a point P in the plane by letting the “x” and “y” coordinates. Suppose the rat also has an unobserved natural tolerance to the poison (Z ). 3. and assume that this tolerance has a standard normal distribution. (a) What is the probability that the rat survives? (b) What is the distribution of Z given that Y = 1? (c) Derive the moment generating function for Z given that Y = 1. Use the moment generating function derived in part (c) to show that φ(−X ) φ(X ) E (Z |Y = 1) = = 1 − Φ(−X ) Φ(X ) . (d) Again. You must justify your answer. level. You might want to express you answer in terms of Φ or φ where φ(·) is the standard Gaussian PDF and Φ(·) is the standard Gaussian CDF. Qy as a function of the coordinates of P and find the joint density function for Qx and Qy . Compute the overall expected number of eggs that will survive as a function of β . 1) random variables. assume that the rat survives. and Z is random. f (x. (c) Are Qx and Qy independent? 2.D. 60 points are needed to pass at the Master’s level and 75 to pass at the Ph. (20 pts) The number of eggs Y laid by an insect has a Poisson distribution with expected value E (Y ) = λ. (c) Suppose that the average number λ of eggs laid by an insect is a function of the age β . assume that λ has an exponential distribution with parameter β = E (λ). (20 pts) A rat is exposed to a known dose of X units of poison and either survives (Y = 1) or dies (Y = 0). Further. You must justify your answer. (a) Compute the overall expected number of eggs that will survive. has a Binomial distribution with sample size y and probability p. y ). be independent N (0. 1. X . Qx . Given the insect lays Y = y eggs. Px and Py . 2002 Work all problems. suppose that the rat survives if and only if Z > −X . (b) Show that V (X ) = E (X ). Express the coordinates of Q. n→∞ lim P (Yn ≤ y ) = Φ(y ) where Φ(x) denotes the cumulative distribution function of the standard normal distribution. . under suitable conditions. . X2 . 2 1 /2 i=1 σi ) It can be shown that. λ {1 µ+ λ − (d) Use parts (a). Let the two times be independent. . We will only observe the time until death (the minimum of the two exponential random variables) and the cause of death. and (c) to determine whether the time until death is independent of the cause of death. Suppose that a computer test can generate an infinite number of questions arranged in a sequence from the easiest to the most difficult. and define Yn = Pn Pn i=1 (Xi ( − µi ) . (b). (a) What is the distribution of the observed time until death? (b) Find the probability that the person dies from cancer. (c) Show that the probability that the person dies of cancer within k months is e−k(µ+λ) }.4. 5. (15 pts) Let X1 . The time until a heart attack is modeled with an exponential distribution with mean µ−1 months. and the time until death from cancer is modeled with an exponential distribution with mean λ−1 months. . Approximate the probability that a student will answer correctly at least 10 out of 100 questions. with E (Xi ) = µi and 2 V (Xi ) = σi . (25 pts) Suppose a person is at risk for two ways of dying. Suppose also that the probability that a student will answer the ith question correctly is pi = 1 i+1 and that all questions will be answered independently. The larger of the two random variables will not be observed. be a sequence of independent random variables. dying from cancer or dying from a heart attack. 01) = . V (Xi ) for 1 ≤ i ≤ n. . 3. . Let α be the area of A. (a) Find the expected value. (c) Compute the variance of Y .99 P (|X when α = . X ¯ is. (c) Find the marginal pdf of Y1 . 2002 Work all problems. level. (24 pts) Let Q be the unit square in the xy-plane and A a region in Q. a good approximation to (b) Let X i=1 n what number? Why? (c) Use the CLT to determine how large n should be so that ¯ − α| ≤ . n. Let Xi = for i = 1. ( 4x1 x2 0 for 0 < x1 < 1 and 0 < x2 < 1 otherwise . E (Xi ). y2 ) such that fY1 . ¯ − α| > .D. .2. ¯ = 1 Pn Xi . (a) Find the probability distribution of Y . with high probability. (18 pts) Let X have pdf f (x) = ( ( 1 if the ith point lies in A 0 otherwise e−x x > 0 0 otherwise and let Y be the greatest integer less than or equal to X . and the variance. (b) Find the joint pdf of Y1 and Y2 . Choose n points independently and unifromly distributed over Q.Y2 (y1 . x2 ) = Let Y1 = X1 /X2 Y2 = X1 ∗ X2 (a) Sketch the region S consisting of all points (y1 .DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS BASIC EXAM – PROBABILITY January 25. 60 points are sufficient to pass at the Master’s level and 75 to pass at the Ph. 1. y2 ) > 0. (b) Compute E (Y ).01) Does your answer (d) Use Chebychev’s inequality to give an upper bound for P (|X require that n be “large?” 2. . If n is large. (18 pts) Let X1 and X2 be random variables with joint pdf f (x1 . and V (Y |X = x) = 1. three of type B. . (12 pts) Three molecules of type A. One such chain molecule is ABCDABCDABCD. . (18 pts) Let X be a standard normal random variable. 6. what is the moment P generating function of Y = n k=1 Xk ? (c) What is the distribution of Y ? 5. What is the probability that all three molecules of each type end up next to each other (as in BBBAAADDDCCC)? e−µ µx x! . three of type C. . Xn are independent Poisson variables with means µ1 . (12 pts) A Poisson random variable with mean µ has pdf f (x) = for x = 0. (c) Show that E (XY ) = a. (b) Show that V (Y ) = 1 + a2 . 2. . It may help to remember that ∞ X yk ey = k=0 k ! (b) If X1 . 1. .4. . µn . and three of type D are to be linked together to form a chain molecule. . for some known a and b. . . (a) How many such chain molecules are there? (b) Suppose all of the different molecule structures are equally likely. and let Y be a random variable such that E (Y |X = x) = ax + b. . . (a) Let X be Possion with mean µ. Compute the moment generating function of X . (a) Show that E (Y ) = b. (a) Derive the moment generating function of X .DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS BASIC EXAM – PROBABILITY August 30. 1. and Bailey’s Fine Fabrics. (a) Find P (B |S c ). For example. . | {z . American Buzz Saw Inc. Find the mean and variance of the number of successful mailings. 60 points are needed to pass at the Master’s level and 75 to pass at the Ph. for example P (A|S ) = 4/5. P (X = x) = for x = 0. he sends a copy to each of the two clients. . (b) Suppose X1 .. (d) Suppose there are n mailings and that the behavior of the system is independent for each item. (b) Compute the variance of T . . however. for the sequence: 11243665 . (17 points) In a children’s game a six-sided die is rolled until all six faces have come up. the letter gets lost with probability 1/4. The letter to ABS has probability 4/5 of being received while the one to BFF has probability 1/6 of being lost. Let S be the event “secretary receives letter.D. If both clients receive their letters. . (16 points) Let X have a Poisson distribution with parameter λ. the president of the agency sends a letter by messenger to her secretary. 2001 Work all problems. . 1. . (17 points) An advertising agency sends out periodic mailings to two clients. Let T be the number of rolls it takes for this to happen. } T =8 (a) Compute the expectation of T . 2.” Thus. XN are independentP random variables with Xi ∼ Poisson(λi ) What is the moment generating function of Y = N i=1 Xi ? (c) What is the distribution of Y ? You must justify your answer. level.” and B be the event “BFF receives letter. (b) Find P (Ac ) and P (S |Ac ). If the secretary receives the letter. For each mailing. 3.. . The letters to the clients. .” A be the event “ABS receives letter. . are received or lost independently of each other. the mailing is deemed successful. e−λ λx x! (c) Determine whether or not the events A and B are independent. if sent. y ) = √ exp { − } 2(1 − ρ2 ) 2π 1 − ρ2 for x. given Y1 = y1 . (17 points) Let X1 and X2 be independent. Let Tn be Joe’s “debt” to Chris after n trips to the BW (Tn could be negative). (d) Do you agree with Joe’s assertion? (c) What is limn→∞ P (|Tn | < c)? (a) What is limn→∞ P (|Tn /n| < c)? ( e−x 0 if x ≥ 0 if x < 0 . (17 points) In this problem you may use any properties of the standard normal density that you need. (b) Find the marginal pdf of Y1 . (16 points) Joe and Chris go the the Blue Wall every day and flip a coin to decide who will buy coffee for the other at a price of $1. y ∈ < where −1 < ρ < 1. identically distributed random variables having common pdf f (x) = Let Y1 = X1 + X2 Y2 = X1 − X2 (a) Find the joint pdf of Y1 and Y2 . (show the computation) (c) Find the marginal distribution of Y .Y) be the random vector whose joint density is f (x. What is the distribution of Y1 ? (c) Find the conditional pdf of Y2 . y ) is a probability density function in the xy -plane. 6.4. Joe says they needn’t keep track of the history since things will “average out” and not matter in the long run. (b) Give a function s(n) so that α = limn→∞ P (|Tn /s(n)| < c) satisfies 0 < α < 1. (b) Let (X. Let 1 x2 − 2ρxy + y 2 f (x. (a) Verify that f (x. For all of the following questions you must justify your answers. for some fixed y1 . Which distribution is it? 5. Compute E (X ). What is α? (your answer needn’t be a number. (d) Find the conditional density of X given Y = y . Let c > 0. y ). but should be computable from a table). (20 pts) Let X1 . Y2 . . and X3 be independent random variables such that Xi has a Gamma distribution with paramters αi . How would the answer change if the first card were the ace of diamonds. (b) Let A1 be the event “first card is an ace. 60 points are needed to pass at the Master’s level and 75 to pass at the Ph. 2001 Work all problems. (d) Returning to the card experiment. 4. Continue this process until no coins are left and let T be the number of the last simultaneous flip. . describe an appropriate sample space for this situation.” and A2 be the event “second card is an ace. Show that P (B |C ) = P (B |C1 ). After the first simultaneous flip remove all the coins which showed heads and flip the remaining coins simultaneously. (20 pts) Consider the following problems involving independent flips of fair coins. Suppose that C1 . . (a) What does the weak law of large numbers tell you about the occurrence of 0’s in the n λαi αi −1 −λi x x e Γ(αi ) . Xi has density fi (x) = 1(0. (b) Find the marginal joint density of Y1 . with P (C ) > 0. That is. . . n. Compute the cdf and pdf of T . let B1 be the event “first card is ace of spaces. 3.” Use the definition of conditional probability to compute P (A2 |A1 ). (20 pts) Two cards are drawn without replacement from a standard deck. Y2 . .D. . with C = ∪k i=1 Ci . (a) Assuming that the order in which the cards are drawn is important. and Xi = 0 otherwise. Let Xi = 1 if the ith draw yields the ball numbered 0.” Compute P (A2 |B1 ). C2 . (15 pts) From an urn containing 10 balls numbered 0 through 9. What is the pdf of N ? (b) Begin by flipping k coins simultaneously. Y2 ) and Y3 are independent. P (Ci ) > 0. (a) Flip a coin until the last two flips are HT and let N be the number of flips required. . . . k . . (c) Consider now an arbitrary sample space Ω and two events C and B in Ω. 1. i = 1.DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS BASIC EXAM – PROBABILITY January 22. Let X1 X1 + X2 + X3 X2 Y2 = X1 + X2 + X3 Y3 = X1 + X2 + X3 Y1 = (a) Find the joint density of Y1 .∞) (x) where αi > 0 and λi > 0. Further suppose that P (B |Ci ) = P (B |C1 ) for i = 2. Ck are disjoint sets. . 2. λi . Y3 . n balls are drawn with replacement. X2 . level. (c) Show that (Y1 . (b) Use the central limit theorem to find an approximate probability √ that. (25 pts) (a) A random variable has a χ2 (ν ) distribution if its pdf is f (x) = with MGF ( 1 xν /2−1 e−x/2 2ν /2 Γ(ν /2) 0 ✓ ◆ x>0 otherwise ν /2 1 MX (t) = 1 − 2t P 2 Let X1 . among the √ n balls thus chosen. the ball numbered 0 will appear between (n − 3 n)/10 and (n + 3 n)/10 if n = 100. . be independent with Xi ∼ χ (νi ) What is the pdf of Y = n i=1 Xi ? (b) A random variable X has moment generating function 1 1 1 MX (t) = et + e2t + e5t 2 4 4 What is the distribution of X ? Why? . . . 5. X2 .