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    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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    Solution Summary

    This solution provides a step-by-step tutorial showing how to compute the given probability questions.