Find the value of the constant c. Find P(YX). Find P(Y = X).

Suppose X and Y are independent. X has a mean of 1 and variance of 1, Y has a mean of 0, and variance of 2. Let S=X+Y, calculate E(S) and Var(S). Let Z=2Y^2+1/2 X+1 calculate E(Z). Hint: for any random variable X, we have Var(X)=E(X-E(X))^2=E(X^2 )-(E(X))^2, you may want to find EY^2  with this. Calculate cov(S,X). Hint: similarly, we have cov(Z,X)=E(ZX)-E(Z)E(X), Calculate cov(Z,X). Are Z and X independent? Are Z and Y independent? Why? What about mean independence?

show all solution 1. Suppose X is a random variable with E(X2)=10 and E(X)=2. Find the following: a) E(2X + 5) b) Var(X) c) Var(2X + 5) d) Var(2X – 5)

X and Y are discrete random variables. If var(X)= var(Y) = o², cov(X,Y)=, find var(2X-3Y)

Suppose X and Y are independent random variables with E(X) =2, E(Y)=3,V(X)=4,V(Y)=16. Finda)E(5X-Y) b)V(5X-Y) c)COV(3X+Y,Y) d)COV(X,5X-Y)

Suppose that X andY are random variables with E(X) = 2 , E(Y) = 5 and E(X²) = 8, E(Y?) = 30 and cov(2X – Y,4X – 3Y) = -12, then cov(-5X – 3, 2Y + 4) is equal to: Hint: cov(aX + br, сх + dr) %3 асV(х) + bdv (Y) + (be + ad)cov(X,Y) -64 -30 -120 -59

Let X1 and X2be independent exponential random variables: fX1(x1) = e−x1 and fX2(x2) = e−x2 1

Suppose X is a random variable such that E(X) = 90 and Var(X) = 7. Compute thefollowing: E(4X + 12) Var(−X) SD(4X)E(X^2) E(X + 3)^2

Suppose X and Y are two random variables with covariance Cov(X, Y) = 3 and Var(X) = 16. Find the correlation coefficient between X and Y.

6) Let X be a random variable that takes values -2, -1, 0, 1, 2; each with probability 1/5. Let YX². Show that Cov(X,Y) = 0 but X and Y are not independent. =