posibility

Math 224 Fall 2017 Homework 4 Drew Armstrong

Problems from 9th edition of Probability and Statistical Inference by Hogg, Tanis and Zim- merman:

• Section 2.3, Exercises 16, 17, 18. • Section 2.4, Exercises 13, 14. • Section 4.1, Exercises 3, 4. • Section 4.2, Exercises 3(a). • Section 5.3, Exercises 2, 5.

Additional Problems.

1. “Collecting Coupons.” Each box of a certain brand of cereal comes with a toy. If there are n possible toys and if they are distributed randomly, how many boxes of cereal do you expect to buy before you get them all?

(a) Let X be a geometric random variable with pmf P (X = k) = p(1 − p)k−1. Use a geometric series to compute the moment generating function:

M(t) = E[etX ] = ∞∑ k=1

etkp(1 − p)k−1 = etp · ∞∑ k=1

[ et(1 − p)

]k−1 = ?

(b) Compute the derivative of M(t) to find the expected value of X:

E[X] = M ′(0) = ?

(b) Assuming that you already have ` of the toys, let X` be the number of boxes of cereal that you buy until you get a new toy. Observe that X` is geometric and use this fact to compute E[X`].

(d) Let X be the number of boxes that you buy until you see all n toys. Then we have

X = X1 + X2 + · · · + Xn. Use this to compute the expected value E[X]. [Hint: See Example 2.5-5 in the textbook for the case n = 6.]