- #1

- 21

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## Homework Statement

Let X

_{1},X

_{2},...,X

_{n}be i.i.d. Normal(μ,σ

^{2}) random variables, where the sample size n≥4. For 2≤k≤n-2, we define:

Xbar = (1/n)SUM(X

_{i}) from i=1 to n

Xbar

_{k}= (1/k)SUMX

_{i}) from i=1 to k

Xbar

_{n-k}= (1/n-k-1)SUMX

_{i}from i=k+1 to n

S

^{2}=(1/n-1)SUM(X

_{i}-Xbar)

^{2}from i=1 to n

S

^{2}

_{k}=(1/k-1)SUM(X

_{i}-Xbar

_{k})

^{2}from i=1 to k

S

^{2}

_{n-k}=(1/n-k-1)SUM(X

_{i}-Xbar

_{n-k})

^{2}from i=k+1 to n

Find distribution of:

a) Xbar

_{k}+ Xbar

_{n-k}.

b) ((k-1)S

^{2}

_{k}+ (n-k-1)S

^{2}

_{n-k})/σ

^{2}

c) S

^{2}

_{k}/S

^{2}

_{n-k}

d) Evaluate E(S

^{2}| Xbar = xbar) Explain.

## The Attempt at a Solution

**I am not asking for someone to do the proof for me(this is a lot of work), but I would love a verbal explanation for what the distributions of each of those might be/why? THANKS!!

a) I know Xbar

_{k}is distributed Norm(μ,σ

^{2}/k) and Xbar

_{n-k}is distributed Norm(u,σ

^{2}/n-k), but I don't know how adding the two distributions together impacts the overall distribution.

b) Same idea where I know I am adding a χ

^{2}distribution with parameter k-1 to a χ

^{2}distribution with parameter n-k-1 all over σ

^{2}, but do not have a deep enough conceptual understanding to comprehend how adding them together affects it.

c) For this one I got as far as finding a χ

^{2}distribution with parameter k-1 divided by χ

^{2}distribution with parameter n-k-1