Proof - expectation of a discrete random variable

Question

A discrete random variable $X >0$

Can you prove : $$\mathbb{E}[X] = \sum_{x>0} \mathbb{P}[ X >= x ]$$

I don't get the question - is it asking for a generic proof or to find an $x$ which satisfies?

Thanks.

Answer 1

It's asking for a generic proof.

Hint: Start with the definition of expectation and assume $X\in \Bbb N^+$. $$\mathsf E(X) = \sum_{x=1}^\infty x~\mathsf P(X=x)$$

$x=\sum_{y=1}^x y

Then show the conclusion also holds for non-integer support intervals.

Answer 2

The right statement should be $EX=\sum_{x=1}^\infty P(X\geq x)$, the index $x$ must belong to $\mathbb N$, not $\mathbb R_+$.

We have

\begin{align*} \sum_{x=1}^\infty P(X\geq x) &=\sum_{x=1}^\infty \sum_{n=x}^\infty P(X=n)\\ &=\sum_{n=1}^\infty \sum_{x=1}^n P(X=n)\\ &=\sum_{n=1}^\infty nP(X=n)=EX. \end{align*}