probabiltiy5

Math 464 - Fall 13 - Homework 6

1. Let X have a Poisson distribution with parameter λ.

(a) Compute the moment generating function of X.

(b) Compute the third moment E[X3] of X.

(c) Let X1, X2, · · · , Xn be independent and identically distributed. Each has a Poisson distribution with parameter λ. Let Z = X1 + X2 + · · · Xn. Use moment generating functions to find the pmf of Z. Note: this should take almost no computation.

2. Let X be a continuous RV with p.d.f.

f(x) =

{

3 4 (1 − x2) if −1 ≤ x ≤ 1

0, elsewhere

Let Y = X4. Find the p.d.f. of Y .

3. Let X be an exponential random variable with parameter λ. Let Y = eX. (a) Whether or not the mean of Y exists depends on λ. Determine the range of λ for which the mean of Y is finite and compute the mean for λ in that range. (b) Find the p.d.f. of Y .

4. A point is chosen at random from a disc of radius 1. We use the uniform distribution on the disc meaning that the probability of a subset of the disc is equal to the area of the subset divided by π, the area of the disc. Let R be the distance from the point to the center of the disc. So R is a continuous random variable with range [0, 1]. Find the pdf of R. Hint: find the cdf first.

5. Let Z be a standard normal RV. (So it has a normal distribution with µ = 0 and σ = 1.) Let X = Z2. Find the pdf of X. Is it in our catalog?

6. Let Z and X be given as in the last problem. The R program on the web generates a bunch of samples of Z and plots a histogram along with the pdf of Z. Note that they agree very well. Write an R program that plots a histogram of X and the pdf you found in the previous problem. (They should agree!)

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7. (Exposition) Let X be a random variable with finite mean and variance. Define a function by

f(t) = E[(X − t)2]

where t is just a real number. So f(t) is a real valued function on the real line. (a) Show that the minimum of f(t) occurs when t is the mean of X. (b) What is the minimum value of f(t)?

For extra fun you can try and answer the same questions for

g(t) = E[|X − t|]

This is not to be turned in, but I am happy to discuss it.

The following problems are worth doing, but are not to be

turned in.

8. Let X be exponential with parameter λ. (a) compute P(X ≥ t) where t ≥ 0. (b) Now let t, s ≥ 0. Compute P(X ≥ t + s|X ≥ s). Compare with your answer to (a). If we think of X as the lifetime of something, e.g., a lightbulb, then P(X ≥ t + s|X ≥ s) is the probabiility the bulb is still burning at time t + s given that it was burning at time s. The relation between your answers to (b) and to (a) is described by saying that the exponential distribution is “memoryless.”

9. The gamma function is defined by

Γ(w) =

∫

∞

0

xw−1 e−x dx

The gamma distribution has range [0, ∞) and depends on two parameters λ > 0, w > 0. The density is

f(x) =

{

λ w

Γ(w) xw−1 e−λx if x ≥ 0

0 otherwise

(a) Show that Γ(w) = (w − 1)Γ(w − 1). Hint: integrate by parts.

(b) Compute the mean and variance of the gamma distribution.

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