# Statistics: Joint, marginal and conditional densities & expectation

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1. Consider the joint pmf p(x,y)=cxy, 1<x<y<3.

a) Find the normalizing constant c.

b) Are X and Y independent? Prove your claim.

c) Find the expectations of X, Y, XY.

2. Conceptual

Suppose (X, Y) have the joint pmf p(x, x+1) = 1/(n+1), x= 0,1,2,.....,n.

a) Are X, Y independent?

b) What is E( YlX = x)?

c) What is Var( YlX=x)?

d) What is Var(X)?

e) What is Var(Y-X)?

3. X, Y are jointly distributed Uniformly within the Unit circle. What is, the joint PDF is

f(x,y) = c, if X2+Y2 < 1

=0, otherwise.

a) Are X,Y independent?

b) Find the marginal PDF of X.

c) Find E(X).

d) Find E(Y).

e) Find E(X-Y).

f) Find Var(X-Y).

4. (expectation of a Quotient) suppose X and Y are independent and X~Be(2,2), Y~Be(3,3). Find E(X2/Y2).

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