Finding $\mathbb{E}[X 1_{X\geq \omega}]$

Question

For all $x$, let $F(x) = 0$ if $ x<1 $, $F(x)=1-1/x^p$ if $ 1\leq x < \omega$ and $F(x) =1$ if $x\geq \omega$, with $0 < p < 1$ and $\omega >1$.

Let $X$ be a random variable with cdf $F$.

Could somebody show me how to compute $\mathbb{E}[X 1_{X\geq \omega}] $?

Answer 1

Hint 1: $$\mathbb{E}[X1_A] = \mathbb{E}[X1_A(\omega)|\omega\in A]\mathbb{P}(\omega\in A)+\mathbb{E}[X1_A(\omega)|\omega\notin A]\mathbb{P}(\omega\notin A)$$ by the law of Total Expectation. Now this can be further simplified by just the definition of indicator variable and then by the given CDF of $X.$