Question

Xsub1 and Xsub2 are independent Exponential ( θ ), θ >0, random variables.

Let S= Xsub1 +Xsub2

A) Write the PDF of S stating any standard results that you use

B) Find E[Xsub1 | S = s] for any given s>0

C) Xsub1 is a so-called unbiased estimator of θ because E[Xsub1] = θ .
Show that E[Xsub1 | S] is also an unbiased estimator of θ AND has a varience no larger than Var(Xsub1)