Homework 10  Due May 2



1. THeorem or Fact: If X is a random variable with CDF F, then E|X-a|= \int |x-a| dF(x), as a function of a,
is minimized at a=median of X. 

Prove this by assuming the X distribution is the following: P(X= c_i) = 1/k, for c_1 < c_2 < ...< c_k.
(please note in the discrete case median may not be unique).

(The theorem is true for discrete and continuous CDF, but continuous CDF can be a bit awkward to handle) 
(you may want to read page 78 prob. 2.17 for the definition of the population median)




2. Suppose P(X=0)=1-p, P(X=1)=p/2, P(X=-1)=p/2, where p is parameter. parameter space 0 <= p <= 1.

 (a) find MLE of p
 (b) show that  T(X) = 2I[X=1] is an unbiased estimator of p
 (c) show that  K(X) = I[X=1  or X=-1]  is also an unbiased estimator of p.  which of the two unbiased estimator has smaller var?