Let X and Y be independent discrete random variables and suppose that X+Y=2. Show that X and Y are constant random variables.

Assume that the random variables X1 and X2 are bivariate normally distributed with mean µ and covariance matrix Σ. Show that if cov(X1, X2) = 0 then X1 and X2 are independent.

Suppose that X and Y are independent random variables with EX=EY=100, and E[min(X,Y))=79. Find ElX-Y|, (the mean of the absolute value of X-Y).

If X and Y are independent random variables with X~N(2,32) and Y~N(1,42),find P(X<Y). (You need to write out the calculation process and figure out theanswer)

Prove that if X and Y are non-negative independent random variables, then X^2 ⊥ Y ^2.

Prove that if X and Y are non-negative independent random variables, then X² 1Y².

Show that if X, Y are independent random variables, then Cov(X, Y ) = 0.

Let X ~ N(1,2) and Y ~ N(4, 7) be independent random variables. Find the probability P(X + Y > 0):

Let X be a random variable and c be a constant. Then Var(X+c) = Var(X), always.   True False

let x be a discrete random variable with probability function p(x)= cx/20 for x = 2,3,4,5,6,8,12 what is the value of c that makes p(x) a valid probability function?