mth 524 hw 8

MTH 524 Section D (Fall 2018) Homework 8

November 5, 2018

1. Let X be a discrete random variable satisfying P(X = k) = 1/3, for k = −1,0,1. Let Y = X2.

Show that X and Y are not independent, however Cov(X,Y ) = 0.

2. Show that if X and Y are independent random variables, then Cov(X + Y,X − Y ) = V (X) − V (Y ). 3. Suppose n balls are drawn one by one at random from an urn with r

red balls and b black balls. For i = 1, . . . ,n, let Xi be the bernoulli random variable that indicates if a red ball was drawn in the ith draw. Calculate Cov(Xi,Xj)

(a) assuming that sampling was done with replacement.

(b) assuming the sampling was done without replacement.

(c) For each case, use the above calculations to calculate the variance of the number of times a red ball is drawn.

4. For a multinomial random vector (X1,X2, . . . ,Xk) with parameters n and (p1,p2, . . . ,pk), use the technique of Bernoulli random variables to calculate Cov(Xi,Xj) for all i and j. Also, calculate ρX1X2.

5. Suppose X!,X2, . . . ,Xn are i.i.d. random variables, each with mean µ and variance σ2, Let S =

∑n k=1 Xk and T =

∑n k=1 kXk. Calculate ρST .

6. In the notation of Problem 5, suppose that U = S + T − ST and that V = (1 − S)(1 − T). Calculate ρUV .

7. Use the technique of moment generating functions to show that the sum of k independent, binomial random variables, each with the same probability of success, is also binomial.

8. Use the technique of moment generating functions to show that the sum of n independent, gamma random variables, each with the same shape parameter λ, is also gamma.

9. Use the moment generating function of the geometric distribution to calculate the expectation and variance.

10. Use the moment generating function to calculate E(Xk), for each positive integer k, if X is a normal random variable with mean 0 and variance σ2.

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