2 Due: Wednesday, 03/01/2023 by 11:59 pm 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 iid 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 iid 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 ����