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 ���.
!! !""