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

A1。如果X是概率密度函数f（x）= 2x的随机变量，则
0 <x <1，则随机变量X的第二个矩为：
（A）1/2；
（B）2/3；
（C）1/3；
（D）4×2
（E）以上都不是。

A2。设X为均值和方差为2的正态分布随机变量

（A）0；
（B）1；
（C）expf0：5g；
（D）expg g;
（E）以上都不是。

A3。在一个人中发现5000的某种癌症。如果一个人确实患有

（A）p =：084;
（B）p =：531；
（C）p =：482；
（D）p =：314；
（E）以上都不是。

B部分

B6。设X f（x;）= x 1

，x> 0，> 0一个未知参数和> 0

（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
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) expfg;
(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).