# if X positive and E(X) < $\infty$ then find a limit

if X a positive random variable and E(X)<$\infty$

how can we find/proof the result of the following limit $\lim_{x\to \infty}$ x P(X> x)

MY THOUGHT: $1)$ I was thinking that we can use something like this if $X>0$ we know $E(X)=\int_0^\infty dx \ P(X>x)$.

$2)$Also I was trying to find something with Dominated Convergence Theorem.

• Welcome to MSE! Can you edit your question to include your thoughts and efforts on this problem? What have you tried, and where are you having difficulty? This will help people write an appropriate answer the addresses your problem. Questions that include this information tend to have a much better response.
– user296602
Commented Jan 5, 2016 at 6:42
• yes sure , i will edit it Commented Jan 5, 2016 at 6:53
• math.stackexchange.com/questions/1595559/…
– user296602
Commented Jan 5, 2016 at 6:54

1.Note $nP(X>n)\to 0$ as $n\to \infty,n\in\mathbb{N}$ iff $xP(X>x)\to 0$ as $x\to \infty,x\in\mathbb{R}$.
2.Let $f_n=n1_{\{X>n\}}$ then $f_n\to 0$ a.e. since $E(X)<\infty$ implies $X<\infty$ a.e.. Also $|f_n|\le X$. Use Dominated convergence theorem.