Math 524 7

MTH 524 Section D (Fall 2018) Homework 7

October 25, 2018

1. For a binomial random variable X, calculate the expectation, the second factorial moment, and the variance directly from the definitions and the frequency function (as we did for the hypergeometric in class)

2. This is a simplified inventory problem. Suppose that it costs c dollars to stock an item (assume the cost is the same regardless of the item being sold or unsold) and that the item sells for s dollars. Suppose that the number of items that will be asked for by customers is a random variable with the frequency function p(k), k = 0, 1, . . . ,. Find a rule for the number of items that should be stocked in order to maximize the expected income.

(Hint: Consider the difference of successive terms.)

3. Find E[1/(X + 1)], where X is a Poisson random variable.

4. Calculate the expectation and variance of a negative binomial random variable by writing it as a sum of geometric random variables.

5. Suppose that n enemy aircraft are shot at simultaneously by m gun- ners, that each gunner selects an aircraft to shoot at independently of the other gunners, and that each gunner hits the selected aircraft with probability p. Find the expected number of aircraft hit by the gunners.

6. Referring to the coupon collection problem discussed in class, what is the expected number of coupons needed to collect r different types, where r < n?

7. A child types the letters Q, W, E, R, T, Y, randomly producing 1000 letters in all. What is the expected number of times that the sequence QQQQ appears, counting overlaps?

8. Suppose X has cdf F (x) = 1 − x−α, x ≥ 1, (α > 0). (a) Calculate E(X) for those values of α that the expectation is defined.

(b) Calculate V (X) for those values of of α that the variance is defined.

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9. Suppose U1, U2, . . . Un are i.i.d. random variables, uniformly dis- tributed on (0, 1).

(a) Calculate the density of U(r) for each r.

(b) Calculate the expectation and variance of U(r).

10. With the set up of the previous problem,

(b) calculate E(U(r) − U(r−1)). (c) calculate E(U(n) − U(1)).

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