STAT 441: Homework 2 Due: Wednesday, 03/01/2023 by 11:59 pm

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STAT 441: Homework 2 Due: Wednesday, 03/01/2023 by 11:59 pm

1. Let ���" be the variance of a random sample of size ��� from a normal distribution where the !!

population variance is ���". Let ���" be the variance of a random sample of size ��� from a !""

normal distribution where the population variance is ���". Assume the two samples are "

  independent. Let

���"/���" ���=!!.

���"/���" ""

 Showthat���~���(���! −1,���" −1).

2. Let ���" be the variance of a random sample of size ���

= 10 taken from a normal population where the population variance is ���". Let ���" be the variance of a random sample of size ��� =

20 taken from a normal population where the population variance is ���". "

If ���" = ���", find the probability ���(���" < ���"). !" !"

3. Let ���(!) ≤ ���(") ≤ ⋯ ≤ ���(%) be the order statistics for a i.i.d. random sample ���!, ���", ... , ���% from an exponential distribution with the parameter ���.

(a) Derive the probability distribution of ���(!). (b) Derive the probability distribution of ���(%).

(c) For random samples of size ��� = 2��� + 1 from this kind of population, derive the probability distribution of the sample median ���:.

4. Let ���!, ���", ... , ���% be a i.i.d. random sample from any population with mean ��� and variance ���". Let ���" be the sample variance of the random sample such that,

∑% (������=)" ���"= &'! &

���−1

Show that ���" is an unbiased estimator of the population variance ���".

(Remark: This is the reason why in ���", the sum of squared is divided by ��� − 1 instead of ���.)

5. Let {���! , ... , ���% } be a random sample of size ��� from an exponential distribution with parameter ���. Show that the sample mean ���= is the minimum variance unbiased estimator (MVUE) of ���.

!! !""

 

 


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