Consider a random variable X that takes only nonnegative integer values. Show that the following
is a way of computing the expectation of X:
E(X) = (sum of this function from x=0 to x= infinity) (sum symbol) Pr(X > x)
Consider a random variable X that takes only nonnegative integer values. Show that the following
is a way of computing the expectation of X:
E(X) = (sum of this function from x=0 to x= infinity) (sum symbol) Pr(X > x)
Well, what is $\displaystyle P(X>x)?$
It is the cumulative distribution function. What do you get when you sum over the distribution function for all $\displaystyle x?$ What do you get when you sum over the distribution function to some $\displaystyle x?$
I'll just make your life easy.
$\displaystyle \sum_{x=0}^{\infty} P(X>x) = \sum_{j=0}^{\infty}\sum_{x=0}^{j} P(X=j) = \sum_{j=0}^{\infty}jP(X=j) = E[X].$
Note this only necessarily holds for integer-valued non-negative random variates.