Re: expected value ques

Posted by Amit Goyal on Jun 08, 2013; 2:50am
URL: http://discussion-forum.276.s1.nabble.com/expected-value-ques-tp7581946p7581973.html

E(N-1) cannot be -1/2 since N-1 always takes non-negative values.
Note that
E(N -1) = E(N) - 1 (by linearity of expectation)
Now let us find E(N) using
E(N) = E(E(N|X))

E(N|X = x) = E(inf{n | Y(n) > x}|X  = x)
               = E(inf{n | Y(n) > x}) (Since X and Y(n) are independent draws)
Note that for 0 < x < 1,
inf{n | Y(n) > x} has an interpretation: the number of trials until first success, where success is defined as Y(n) > x. And the probability of success in each independent trial is (1-x).  
E(N|X = x)
= E(inf{n | Y(n) > x})
= 1(1-x) + 2x(1-x) + 3x^2(1-x) +......
= 1/(1-x)

Now coming back to E(N):
E(N) = E(E(N|X)) = E(1/(1-X)) = ∞ (infinity)
Thus,
E(N-1) = ∞.