1.(a) Suppose that X1; X2; X3; X4 are independent random observations drawn from a Pois
son distribution with mean λ = 1/4. Let X denote the sample mean:

Calculate P （X < 1/ 2）. Show your working.

(b) Suppose that X has a Poisson distribution with mean λt. Let Y ∼ Gamma(k; λ),where k is a positive integer.

By showing that both the left side and the right side of the equation below can be regarded as the probability of the same event in a Poisson process in which the expected number of occurrences per unit of time is λ, explain why the following identity is true:

P(X ≥ k) = P(Y ≤ t)

2. Let X ∼ Exponential(1). Consider the random variable S, which is independent of X,and has the following property:

S =  1 with probability 1

−1 with probability 1= =2 2

Let Y = SX.

(a) Let y be a real number. Fill in the gaps indicated with : : : in the following identity:

Name the theorem that is the basis for the identity above. [2 marks]

(b) Find the cdf of Y . Show your working. [6 marks]

(c) Find the pdf of Y . Show your working. [2 marks]

(d) Let V = pX. Find the pdf of V , remembering to include the range of values of v for which your answer is valid. [5 marks][Total: 15 marks]

3. Let X1; X2; : : : ; Xn (n > 1) be iid random variables having the Normal distribution with
mean µ and variance σ2. We define:

S = X2 + X3 + · · · + Xn;
X = X1 + X2 + · · · + Xn / n: