这是一个新西兰的统计代写限时测试
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.
Answer the following questions:
(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:
Answer the following questions:
(a) Find the distribution of S, with parameters.