1.(a) 假设 X1;X2;X3;X4 是从 Pois-

4
.让 X 表示样本均值：
X = X1 + X2 + X3 + X4

.展示你的工作。 [3 分]

(b) 假设 X 具有均值为 t 的泊松分布。让 Y Gamma(k; ),

P(X k) = P(Y t)
[3 分]
【总分：6分】

2. 设 X 指数(1)。考虑与 X 无关的随机变量 S，

S =

1 概率 1=2
1 概率 1=2

(a) 设 y 是一个实数。用 :: 表示的空白处填写如下：
P(Y y) = P(SX yj : : :)P(: : 🙂 + P(SX yj : : :)P(: : :):

(b) 找出 Y 的 cdf。展示你的工作。 [6 分]
(c) 求 Y 的 pdf。展示你的工作。 [2 分]
(d) 令 V =

X. 找到 V 的 pdf，记住包括 v 的值范围

[总分：15 分]

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

. Show your working. [3 marks]

(b) Suppose that X has a Poisson distribution with mean t. Let Y  Gamma(k; ),
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 , explain why the following identity is true:
P(X  k) = P(Y  t)
[3 marks]
[Total: 6 marks]

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=2
1 with probability 1=2
Let Y = SX.
(a) Let y be a real number. Fill in the gaps indicated with : : : in the following identity:
P(Y  y) = P(SX  yj : : :)P(: : 🙂 + P(SX  yj : : :)P(: : :):
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 =
p
X. Find the pdf of V , remembering to include the range of values of v for