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这是一篇是完成统计概率的测试题
MAST20006 Probability for Statistics /MAST90057 Elements of Probability
Assignment 5, Semester 1 2020
1.设X1，X2，…。 。 。 ，Xn是大小的随机样本（即X1，X2，…，Xn是独立的）
n来自成功概率p = 0.8的几何分布。
（a）查找mgf MY4
使用几何mgf，Y4的（t）= X1 + X2 + X3 + X4。然后
命名Y4的分布。
（b）查找mgf MYn
Yn的（t）= X1 + X2 +··+ Xn。然后命名分布
恩
（c）查找mgf MY
（t）样本均值Y
n =
n

。
（d）求极限limn→∞MYˉn
（t）使用（c）的结果。什么分布
极限mgf对应于
2
3
（e）让
锌=
√

ˉ
n
5
4
√
5
4
！
=
4
5
√
5n Y
n
√
5n。
寻找锰锌
（t），锌的mgf。然后使用Maple或理论参数来查找
极限mgf limn→∞MZn
（t）。锌的极限分布是什么
4
2.令X1，X2和X3为指数分布中大小为n = 3的随机样本
pdf f（x）= 1
2

-x / 2
，0 2) Keep 3 significant digits after the decimal point in your
result.
5. Let X be a continuous random variable with the density function
f(x) =

2
x
2
, 1 < x < 2 0 elsewhere (a) Find the kth moment of X for k ≥ 1. 20 (b) Find the cdf of X and explain how you could use it to simulate a realisation/observation from X. 21 (c) Generate a sample of 1000 observations from the distribution of X and plot the corresponding histogram. Compare the shape of the histogram with that of the density of X. For this question, sketch the graph of the density and write your comment on the comparison of this graph with the histogram in the box below; refer to your attached code for the histogram. (d) Let Z = ln(X). Find the mgf MZ(t) of Z. For which values of t is it well defined Compare its expression with your answer to (b) and comment. 22 (e) Find E[Z]. 6. Let X be a Gamma distribution with mean 8 and variance 32. Let X1, X2, . . . , Xn be n independent random variables with the same distribution as X. Let Yn = Pn i=1 Xi/n be the sample mean. Sketch the graphs of the pdf of X and the pdf of Yn for n = 3, 5. What do you observe Can you compare Yn to a known distribution when n is large Elaborate on your answer. 23