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?
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 ).
1. Suppose that the joint density of X and Y is given by exp(-y) (1- exp(-x)), if 0 S y,0 syS oo exp(-x) (1- exp(-y)), if 0SyS ,0 oo (e,y)exp(-y) Then . The marginal density of X (and also that of Y), ·The conditional density of Y given X = x and vice versa, Cov(X, Y) . Are X and Y independent? Explain with proper justification.