Description
1. (10 marks) A die is rolled twice. Find the joint probability mass function of X and
Y if X denotes the value on the first roll and Y denotes the minimum of the values
of the two rolls.
2. (15 marks) Let X and Y be independent random variables both following b(n,p).
(a) Compute the covariance of X + Y and X − Y.
(b) Are X + Y and X − Y independent? Please explain.
3. (15 marks) Let X and Y have the joint p.d.f.
f (x, y) = 2(x + y), 0 < x < y <1.
Find the marginal p.d.f. of X and the marginal p.d.f. of Y. Determine whether X
and Y are independent.
4. (15 marks) The joint p.d.f. of X and Y is given by
f (x, y) = exp(−(x + y)), 0 ≤ x < ∞,0 ≤ y < ∞
Find P(X+Y>2).
5. (15 marks) Suppose that the joint p.d.f. of X and Y is
f (x, y) = 6y, 0 < y < x <1.
Find the conditional mean and conditional variance of Y given X=0.3.
6. (15 marks) Using moment generating function, show that if X and Y both follow
Geometric distribution with parameter p, X+Y follows a negative binomial
distribution. Assume that X and Y are independent.
7. (15 marks) Let X and Y have the joint p.m.f. described in the table:
(x,y) (1,1) (2,1) (3,1) (1,2) (1,3) (2,2)
f(x,y) 1/15 4/15 4/15 1/15 2/15 3/15
Calculate the correlation coefficient of X and Y.
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