True or false about the expected value [duplicate]

Question

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• Show that $E(X)=\int_{0}^{\infty}P(X\ge x)\,dx$ for non-negative random variable $X$ 3 answers

Let $X$ be a discrete random variable, then $\mathbb{E}[X]=\sum_{n=1}^{\infty}\mathbb{P}(X \geq n)$

Is this true or false ?

I think it's false , just by using the definition of $\mathbb{E}[X]=\sum_{n=1}^{\infty}x_n\mathbb{P}(X=x_n)$. But I'm not so sure, it seems to be so trivial. I'd like to find a counter-example

Answer 1

It is true under the extra condition that $\Pr(X\in\{0,1,2,\dots\})=1$.

If in that case $p_n:=\Pr(X=n)$ then: $$\mathbb EX=p_1+2p_2+3p_3+\cdots=(p_1+p_2+p_3+\cdots)+(p_2+p_3+\cdots)+(p_3+\cdots)+\cdots$$