Let the p.m.f. (probability mass function) of X be de…noted by
f(x) = 1 / x(x + 1) ; for x = 1, 2, 3,...
1. Show that f is a p.m.f.
2. What is E [X] in this case?
Any help will be greatly appreciated!!!
Let the p.m.f. (probability mass function) of X be de…noted by
f(x) = 1 / x(x + 1) ; for x = 1, 2, 3,...
1. Show that f is a p.m.f.
2. What is E [X] in this case?
Any help will be greatly appreciated!!!
you need to show that this sums to one, and it does
use partial sums to obtain
$\displaystyle {1\over x(x+1)} ={1\over x}-{1\over x+1}$
So $\displaystyle \sum_{x=1}^N {1\over x(x+1)} =\sum_{x=1}^N {1\over x} -{1\over x+1}= 1-{1\over N+1}\to 1 $
as $\displaystyle N\to\infty $ since this is a telescoping series
The mean is infinite since that is a p-series (p=1)
So $\displaystyle E(X)=\sum_{x=1}^{\infty} {x\over x(x+1)} =\sum_{x=1}^{\infty} {1\over x+1}=\infty $
YOU can use the integral test and obtain $\displaystyle \log N\to\infty$
but that's where the p-series comes from.