A random variable N takes integer values 1,2,3,..., where P(N=n)=1/2^n . Find E(N)

Question

A random variable $N$ takes positive integer values $1,2,3,...$ where $P(N=n)=\frac{1}{2^n}$.

Find $E(N)$, where $E(N)= \sum_{n=1}^\infty n P(N=n) $.

Answer 1

Hint:

use $$1+x+x^2+...={1\over 1-x}$$

and then take the derivate of this sum, so you get:

$$1+2x+3x^2 +4x^3+ ...= {1\over (1-x)^2}$$

Now: $$x+2x^2+3x^3 +4x^4+... = {x\over (1-x)^2}$$

Now put $x=1/2$...