How to calculate conditional expectation $E[X|X \geq 0]$?

Question

I only find some materials relating to conditional expectation like $E[X|Y=y]$. How to calculate conditional expectation $E[X|X \geq 0]$?

Answer 1

Let $Y = I(X \ge 0)$. Then, the density of $Y$ is given by:

$$\frac{f(y)}{\int_0^{\infty}f(y) dy}$$

You can then compute $E(X|X\ge 0)$ by computing $E(Y)$.

Answer 2

In general: if $A$ is an event with positive probability, then $$E[X | A] = \frac{E[X \cdot 1_{A}]}{P(A)}$$ where the symbol $X \cdot 1_A$ denotes the random variable that is $X$ on the event $A$ and is $0$ otherwise.

One fact that may be helpful: if $A$ and $B$ are a partition of $\Omega$ (that is, $A \cap B = \emptyset$ and $A \cup B = \Omega$) then: \begin{align*} E[X] &= E[X \cdot 1_A + X \cdot 1_B] \\ &= E[X \cdot 1_A] + E[X \cdot 1_B] \\ &= E[X | A] P(A) + E[X | B] P(B). \end{align*}