APPM 5720 Problem Set One (Due Wednesday, September 13th)

Welcome to Problem Set One! Note that we may not have talked about every distribution that comes up in the assignment. Use your table of distributions!

1. Compute the mean of the Γ(α, β) distribution by “integrating without integrating”.

2. Consider a sequence of independent trials of an experiment where each trial can result in one of two possible outcomes, “Success” or “Failure”. Suppose that the probability of success on any one trial is p. Let X be the number of trials until the rth success is observed, where r ≥ 1 is an integer.

(a) Derive the pdf for X. Show your work. (b) Name the distribution by matching your resulting pdf (pmf) up with one in the table of distributions. (c) Find the distribution of Y = X −r. (Name it!) What is the interpretation of Y in terms of the success/failure experiment? (d) Give an alternative name for the distribution from part (b) in the case when r = 1.

3. (a) Let X ∼ unif(0, 1). Find the distribution of Y = − ln X. (Name it!) (b) Let X be uniformly distributed on the from 0 to 1 (ie: X ∼ unif(0, 1)). Find the transformation y = g(x) such that Y = g(X) ∼ exp(rate = λ).

4. Suppose that X has a (continuous) Pareto distribution with parameter γ. (i.e.: X ∼ P areto(γ)) Find the distribution of Y = ln(X + 1). (Name it!)

5. Suppose that X is a continuous with cdf F . Assume that X is invertible. Sup- pose also that U has a uniform distribution on the interval (0, 1). (We write U ∼ unif(0, 1).) Show that F −1(U) has the same distribution as X. We write F −1(U) =d X to say that they are “equal in distribution”.

6. Suppose that X is a continuous random variable with pdf f(x). Let Y = X2. (Note that this is not a one-to-one, invertible transformation.) Find an expression for the pdf of Y in terms of the pdf of X.

7. Suppose that X1 and X2 are independent random variables each with the 2 iid with mean µ = 0 and σ = 1. We write X1,X2 ∼ N(0, 1). Find the distribution of X1/X2. (Name it!) iid 8. Suppose that X1,X2 ∼ N(0, 1).

Show that Y1 := X1 + X2 and Y2 := X1 − X2 are independent.

9. Let U1 and U2 be independent unif(0, 1) random variables.

Show that X1 and X2 defined as √ X1 = −2 ln U1 cos(2πU2) √ X2 = −2 ln U1 sin(2πU2)

are independent standard normal random variables.

10. Suppose that X1,X2,...,Xn is a random sample from the Pareto distribution with parameter γ. Find the distribution of X(1) = min(X1,X2,...,Xn). (Name it!)

11. Let X1,X2,...,Xn be a random sample from the uniform distribution over the interval (0, 1). Find the of X(n), where X(n) is the maximum of the X’s.

12. Let X1,X2,...,Xn be a random sample from the with rate λ. Let Y1 = X(1) be the minimum value of that sample. Similarly, let Y2 be the minimum of an independent random sample of size n from this distribution, let Y(3) be a minimum of another independent random sample of size n, etc...

Consider Y1,Y2,...,Ym, the minimums for each of m indepenent random samples, each of size n, from the exponential distribution with rate λ. Let Y(1) = min(Y1,Y2,...,Ym). Find P (Y(1) > x).