
7. (EXTRA CREDIT) Suppose the X ~ Exp(1) and Y ~ Exp(1) are independent. Let Z...
3-5.2. Let X, Y, and Z have the joint pdf 3/2 1 |ryz exp exp 27T 2 where -o<x < o,-00< y < oo, and 00< z < 00. While X, Y, and Z are obviously dependent, show that X, Y, and Z are pairwise independent and that each pair has a bivariate normal distribution
3-5.2. Let X, Y, and Z have the joint pdf 3/2 1 |ryz exp exp 27T 2 where -o
7. Suppose that Xi,..., Xk are independent random variables, and X, ~ Exp(B) for i = 1, . . . , k. Let Y = min(X1 , . . . , Xk). Show that Y ~ Exp(Σ-1 β).
Let X and Y be independent variables with X ~ EXP(mu(x)) and Y ~ EXP (mu(y)), where mu(x) = 1 and mu(y) = 1/2. Write explicit integral expressions for each of the following, without computing the values. P(Y < X)
Extra: Let X, Y, Z be results of three independent tosses of a fair die. (a) Find the covariance of the random variables W=2X-3Y + Z (b) Find the correlation coefficient of W and V. and V=X-2Y-Z
Question 2 (5101) Suppose that X and Y are independent, and that Z = X+Y. If X Exp(B = 1) and Y~ Unif(-1,1], what is the density of Z?
Question 2 (5101) Suppose that X and Y are independent, and that Z = X+Y. If X ~ Exp(B = 1) and Y~ Unif(-1,1], what is the density of Z?
Let X1~ exp(1) and X2 ~ exp(1) be independent and identically-distributed exponential random variables with rate 1. Let: Y = X1 + X2 , Z = X1 − X2 (a) What is the cdf of X1? (b) What is the joint pdf of (X1, X2)? (c) What is the joint pdf of (Y, Z)? (d) What is the marginal pdf of Z?
Problem 5 Suppose X and Y are independent random variables following Uniform[0, 1]. Let Z- (X +Y)/2. (1) Calculate the cumulative density of z. (2) Calculate the density of Z.
Problem 5 Suppose X and Y are independent random variables following Uniform[0, 1]. Let Z- (X +Y)/2. (1) Calculate the cumulative density of z. (2) Calculate the density of Z.
A joint pdf is given as Let Z = X/(Y-X), determine if Y and Z are independent random vars f(x, y) exp(-y),0 <y, >0 f(x, y) exp(-y),0 0
Let X and Y be independent random variables with X = N(0, 1) and Y = Exp(1). Find E( |X| (Y + 1)^2 ).