问题 问答题

设总体X的概率密度为


其中θ是未知参数.
(Ⅰ) 求θ的矩估计量,判断它是否为θ的无偏估计量;
(Ⅱ) 求θ的最大似然估计量,判断它是否为θ的无偏估计量.

答案

参考答案:设X1,X2,…,Xn是来自总体X的样本,x1,x2,…,xn是样本值,[*]是样本均值.
(Ⅰ)由于
[*]
令t=x-θ,得
[*]
[*]
得θ的矩估计量为
[*]
因为
[*]
所以[*]是θ的无偏估计量.
(Ⅱ)似然函数为
[*]
当xi≥θ(i=1,2,…,n)时,L(θ)>0,且θ越大,L(θ)越大,取[*]=min(x1,x2,…,xn),有L([*])≥L(θ).θ的最大似然估计量为
[*]
X的分布函数为
[*]
[*]=min(X1,X2,…,Xn)的分布函数为
[*]
[*]的概率密度为
[*]
由此可得
[*]
因为[*],所以[*]=min(X1,X2,…,Xn)不是θ的无偏估计量.

解析:

[分析]: 容易求解(Ⅰ).根据似然函数表达式寻找其最大值点,作为θ的最大似然估计.求出该估计量的分布函数,进而求得其概率密度及数学期望,判断其是否为θ的无偏估计量.

阅读理解

It is that time of year again. Your child’s birthday is just around the corner. There is so much to do to get ready for the party. Planning a child’s birthday party is necessary. It can be fun, exciting and sometimes busy. When planning a child’s birthday party, there are many important details( 细节) to remember:

First, you should decide on a theme. It is very important to plan a successful party. Without a theme there would be no party to plan. If the child is old enough, he can choose what theme he wants to have. He can also choose invitations, decorations, cake design, and even games.

Next it is time to decide on a date, a place, and the time to have the party . Most people have the child’s party on the weekend for conveniece (方便) of the parents’ schedule (日程安排). When you have already decided on a date and time, you need to choose a place to have the party. It should be a place your child will enjoy, such as a favorite park, a nice restaurant or your own home.

Another important detail to keep in mind is the guest (客人) list. Always be sure to only invite people that your child knows and likes. Let the child help you write out the guest list. This helps avoid (避免) unwanted guests.

小题1:The underlined sentence means “________”

A.Your child’s birthday is coming soon.

B.Your child will have a wonderful birthday.

C.You can buy your child gifts in the shop around the corner.

D.Your child would like to have his birthday party around the corner.小题2:The writer advises us to decide on_______ first for a birthday party.

A.a plae

B.a date

C.a cake

D.a theme小题3:The writer advises us to _______.

A.make a guest list for the party

B.invite very important people

C.have the party in the evening

D.invite as many guests as we can小题4:The passage is written to __________.

A.parents

B.teachers

C.children

D.guests小题5:The passage mainly tells us_________.

A.when and where to have a birthday party

B.how to plan a child’s birthday party

C.why a child’s birthday party is necessary

D.what a child should do on his birthday party

单项选择题