probability2

Math 464 - Fall 11 - Homework 3

1. The probability mass function of a discrete RV X is given in the table below. Compute the following:

(a) the probability X is even (b) the probability that 1 ≤ X ≤ 8 (c) the probability that X is −4 given that X ≤ 0. (d) the probability that X ≥ 3 given that X > 0.

x -4 -1 0 2 4 5 6 f(x) 0.15 0.2 0.10 0.1 0.2 0.2 0.05

2. Compute the mean and variance of the geometric distribution. Hints: It is sometimes easier to compute E[X(X −1)] than E[X2]. Note that E[X(X − 1)] = E[X2] − E[X]. The geometric series formula is

∞ ∑

n=0

r n =

1

1 − r

If you differentiate this once and then again you get the useful identities:

∞ ∑

n=1

nr n−1 =

d

dr

1

1 − r ,

∞ ∑

n=2

n(n − 1)rn−2 = d2

dr2 1

1 − r

The following is not to be turned in. This same “summation by differen- tiating” trick can be used in other setting. Take the binomial identity

(x + y)n = n

∑

k=0

(

n

k

)

x k y n−k

Differentiate with respect to x, then set x = p, y = 1 − p and you get an identity that helps you compute the mean of the binomial distribution. Differentiate twice and you get an identity that will lead to the variance.

3. (Exposition) Let X be a discrete RV whose range is 0, 1, 2, 3, · · ·. Prove that

E[X] = ∞ ∑

k=0

P(X > k)

1

4. (Exposition) Let X be the number of emails that a company receives in a day. Assume that X is a Poisson random variable with parameter λ. The company classifies each email as spam or not spam. The probabilty that a single email is spam is p. Let Y be the number of spam emails that the company receives in a day. Assume that for any two emails, whether or not they are spam are independent events. (a) Find E[Y ]. (b) Show that Y is also a Poisson random variable and find its parameter (in terms of λ and p). Hint: Use the partition theorem:

P(Y = k) = ∞ ∑

n=k

P(Y = k|X = n)P(X = n)

If we are given that X = n, then Y is a binomial RV.

5. A coin has probability p of heads. We flip it until we get 3 heads in a row. Let X be the total number of flips. Find the expected value of X. Hint: The idea is to use the partition theorem and the fact that whenever you get a T, things start over. In class we looked at the probability of a run of 3 heads when you flip the coin a fixed number of times. The partition that was useful there will be useful here.

6. Let X be a discrete RV with the geometric distribution. (a) Compute P(X > n). (b) Show that P(X > n + k|X > n) = P(X > k). Because of this result it is often said that the geometric distribution is “memoryless.” (c) (Exposition) The geometric distribution may be realized by flipping a biased coin until we get heads. (X is the total number of flips needed.) Explain why the equation in part (b) is “obvious” in this experiment.

7. The experiment is that we roll a six-sided die ten times. We let X be the sum of the ten rolls. We want to compute E[X|X > 30]. In principle we can compute this analytically, but it will be a real mess. So the point of this problem is to estimate it by simulation, i.e., write an R program to estimate it. To get you started, the R program on the web does the experiment of simulating a sequence of ten rolls and computing X. It does the experiment 20 times and prints out both the sequences of 10 rolls and their sums.

2