How is $E(X;X\ge x)$ defined?

Question

I came across the formula $E(X;X\ge x)=P(X\ge x)E(X\mid X\ge x)$. How do I interpret the LHS? I can understand conditional expectation. I do not know how the LHS is defined.

Answer 1

In general, if $A$ is some event, and $X$ is a random variable, then

$E(X;A) = E(X\boldsymbol{1}_A),$

where $\boldsymbol{1}_A$ is the indicator function on the set $A$. In your case, you have that

$E(X;X\geq x) = E(X\boldsymbol{1}_{\{X\geq x\}}).$