Let X1,X2,...,Xn be i.i.d. Normal(μ,σ2) random variables, where the sample size n≥4. For 2≤k≤n-2, we define:
Xbar = (1/n)SUM(Xi) from i=1 to n
Xbark = (1/k)SUMXi) from i=1 to k
Xbarn-k = (1/n-k-1)SUMXi from i=k+1 to n
S2=(1/n-1)SUM(Xi-Xbar)2 from i=1 to n
S2k=(1/k-1)SUM(Xi-Xbark)2 from i=1 to k
S2n-k=(1/n-k-1)SUM(Xi-Xbarn-k)2 from i=k+1 to n
Find distribution of:
a) Xbark + Xbarn-k.
b) ((k-1)S2k + (n-k-1)S2n-k)/σ2
d) Evaluate E(S2 | 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 Xbark is distributed Norm(μ,σ2/k) and Xbarn-k is distributed Norm(u,σ2/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 χ2 distribution with parameter k-1 to a χ2 distribution with parameter n-k-1 all over σ2, 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 χ2 distribution with parameter k-1 divided by χ2 distribution with parameter n-k-1