本次英国代写主要为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 <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
:
随机变量expf(X)= g的期望由下式给出:
(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
疾病,在92%的情况下,诊断程序会表明他或她
实际上有它。如果一个人没有疾病,诊断程序
每500例中有1例给出假阳性结果。的概率
测试结果呈阳性的人患有癌症(至小数点后三位):
(A)p =:084;
(B)p =:531;
(C)p =:482;
(D)p =:314;
(E)以上都不是。
B部分
在B部分中,每个问题最多可打25分。回答
提供的手册中的所有问题。
B6。设X f(x;)= x 1
设
,x> 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).