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这是一篇英国作业主要为Probability and Statistics的限时测试
A节
A节中的所有问题均带有相同的分数。每个问题
A部分带有10个正确答案的分数，-1代表错误的答案
答案，0表示没有答案，或者选择了多个答案。
在提供的答案网格中将答案写到A部分。每个
问题有一个正确的选择。正确的选项必须是
在网格上指示。
A1。如果X是概率密度函数f（x）= 2x的随机变量，则
0 0，> 0一个未知参数和> 0
已知参数。让X1; ：Xn是随机样本。
（a）找到最大似然估计^ MLE。
（b）检查^ MLE是否为的无偏估计。
（c）根据（b）部分的结果，指示的无偏估计量。
（d）找到Cramer-Rao下界。
（e）比较Cramer-Rao下界与无偏es-的方差
（c）的刺激者。
A jié
SECTION A
All questions in Section A carry equal marks. Each question in
Section A carries 10 marks for the correct answer, -1 for a wrong
answer, 0 marks for no answer or if more than one answer is chosen.
Write your answers to Section A in the answer grid provided. Each
question has exactly one correct option. The correct option must be
indicated on the grid.
A1. If X is a random variable with probability density function f(x) = 2x, for
0 < x < 1, then the second moment of the random variable X is: (A) 1/2; (B) 2/3; (C) 1/3; (D) 4×2 (E) None of the above. A2. Let X be normally distributed random variable with mean and variance 2 : The expectation of the random variable expf(X )=g is given by: (A) 0; (B) 1; (C) expf0:5g; (D) expfg; (E) None of the above. A3. A certain cancer is found in one person in 5000. If a person does have the disease, in 92% of the cases the diagnostic procedure will show that he or she actually has it. If a person does not have the disease, the diagnostic procedure in one out of 500 cases gives a false positive result. The probability that a person with a positive test result has the cancer is (to 3 decimal places): (A) p = :084; (B) p = :531; (C) p = :482; (D) p = :314; (E) None of the above. SECTION B In Section B each question carries a maximum of 25 marks. Answer all questions in the booklet provided. B6. Let X f(x; ) = x 1 e x , x > 0, > 0 an unknown parameter and > 0
a known parameter. Let X1; : : : ;Xn be a random sample.
(a) Find the maximum likelihood estimator ^ MLE of .
(b) Check if ^ MLE is an unbiased estimator of .
(c) On the basis of the result of part (b), indicate an unbiased estimator of .
(d) Find the Cramer-Rao lower bound.
(e) Compare the Cramer-Rao lower bound to the variance of the unbiased es-
timator of part (c).