STATISTICS 251 HOMEWORK ASSIGNMENT 8 DUE FRIDAY NOV 18 Problem 1. Let U1 , U2 be independent uniform random variables on [0, 1], and write M = max(U1 , U2 ). (a) Find the conditional joint density of (U1 , U2 ) given M 1/2. (b) Find the conditional density of U1 given M 1/2. Problem 2. Let X, Y, Z be independent random variables where X, Y are uniformly distributed on [0, 1] and Z is an exponential with mean 1. (a) What is the conditional density of X given XY = t for 0 t 1? (b) What is the conditional density of X given X + Z = t for t 0? Problem 3. Let X1 , . . . , Xn be i.i.d. standard Gaussian variables, and let Sk = 1, . . . , n. Let m < n be an integer. (a) Find the conditional distribution of Sn given Sm = s. (b) Find the conditional distribution of Sm given Sn = t. Pk i=1 Xi for k = Problem 4. Let X1 , . . . , X6 be independent standard Gaussian variables. Compute the probability that P(X12 + X22 (X3 + X4 )2 + (X5 + X6 )2 ). (Hint: we have found the distribution of X12 + X22 in previous homework assignment) Problem 5. Suppose that in a Marathon, the runners arrive at the destination as a Poisson process with rate 1 between time 0 and 10. Suppose that every runner who arrives at time s get a reward of (10 s) dollars, for all 0 s 10. Denote by M the total amount of dollars that have been given to runners who arrive at a time between 0 and 10. (a) Compute P(M = 0). (b) Compute EM . (c) Compute VarM . Problem 6. Independent Gaussians. Suppose that X and Y are independent random variables each with the standard normal distribution. Let X = R cos and Y = R sin be the polar coordinate representation of the point (X, Y ), with the angular coordinate chosen so that 0 < 2. (A) Find the density of Y /X. (Hint: connect this to the angle of the point (X, Y )). = R cos 2 and Y = R sin 2 are independent standard normal random variables. (B) Show that X (C) Use (B) to show that the random variables 2XY X2 + Y 2 and X2 Y 2 X2 + Y 2 are independent standard normal random variables. N OTE : You can either look up the relevant trig double-angle formulas or you can forget all the trig you ever knew and just learn Euler's formula ei = cos + i sin . 1 Problem 7. Addition of Gammas. The Gamma density with shape parameter > 0 and scale parameter > 0 is f, (x) = x1 ex () for x 0, = 0 for x < 0. The constant () is a normalizing constant. (When is a positive integer, () = ( 1)!.) (A) Show that if X and Y are independent random variables with densities f, and f, , respectively, then X + Y has density f+, . N OTE : We did the case where , are integers in class. (B) Show that if X is standard normal then X 2 has density f1/2,1/2 . Pn (C) Show that if X1 , X2 , . . . , Xn are independent standard normal then i=1 Xi2 has density fn/2,1/2