1. Toss a biased coin (with probability p of heads and 1p of tails) until the first head appears. Let N be the number of tosses required to get the first head. Now toss a fair die N times, and let S be the sum of results of the N tosses of the die. Use conditioning to compute E[S] and Var[S] under the following circumstances:

(a) The die has every face numbered 3.

(b) The die has the faces numbered 1, 2, 3, 4, 5, 6.

2.Let X and Y be independent random variables. Show that g(X) and h(Y ) are also independent, where g and h are functions from R to R.

3.Let X₁ and X₂ independent geometric random variables with parameters p₁ and p₂ respectively.

(a) Find P(X₁ X₂).

(b) Find P(X₁ = X₂).

4. Let X₁ and X₂ be independent geometric random variables with parameters p1 and p2 respectively. Let D = X₁ X₂ and M = min(X₁, X₂).

(a)Find the joint PMF of D and M.

(b) Compute the marginal PMFs of D and of M.

(c) Are D and M independent? Explain.

5. Let X₁ and X₂ be independent Poisson random variables with the same parameter . What is the distribution of S = X₁ + X₂? This is Possion Compounding, the opposite of Poisson Thinning.

6. Let X₁ and X₂ be independent random variables that are uniformly distributed on {1, ..., n}. What is the PMF of S = X₁ + X₂?