What is $E(X\mid X>c)$ in terms of $P(X>c)$?
I've seen conditional probability/expectation before with respect to another random variable but not to the variable itself. How would I go about interpreting this?
What is $E(X\mid X>c)$ in terms of $P(X>c)$?
I've seen conditional probability/expectation before with respect to another random variable but not to the variable itself. How would I go about interpreting this?
$$ \mathrm E(X\mid X\gt c)=\frac{\mathrm E(X\cdot\mathbf 1_{X\gt c})}{\mathrm P(X\gt c)} $$
@Dilip has already provided an example of how to compute $E(X|X>c)$ in the discrete case. In the continuous case, the computation will be:
$$E(X|X>c) = \frac{\int_x x f(x) I(x>c) dx}{\int_x f(x) I(x>c) dx}$$
Thus, if we let $X \sim U[0,1]$ and $c=0.5$ then we have:
$$E(X|X>0.5) = \frac{\int_x x I(x>0.5) dx}{\int_x I(x>0.5) dx} = \frac{(1 - 0.5^2)/2}{0.5}=0.75$$
In terms of interpretation: Intuitively, you can think of $E(X|X>c)$ as the average of all $X$ values subject to each one of them being greater than $c$.