Let X be an non-negative random variable. Can any one tell me how to prove the following inequality?
$\mathbb{P}(X>\lambda_0)\cdot \mathbb{E}[X|X>\lambda_0]\leq \int_{\lambda_0}^{\infty}\lambda\cdot \mathbb{P}(X>\lambda)d\lambda$.
$\lambda_0$ is a positive number.
I am very confused about the conditional expectation here. Is my following calculation true?
$\mathbb{E}[X|X>\lambda_0]=\int_0^{\infty}\frac{\mathbb{P}(X>\lambda|X>\lambda_0)}{\mathbb{P}(X>\lambda_0)}d\lambda=\int_0^{\lambda_0}d\lambda+\int_{\lambda_0}^{\infty}\frac{\mathbb{P}(X>\lambda)}{\mathbb{P}(X>\lambda_0)}d\lambda$.
Many thanks in advance!