If I have a random variable $X$ then how to interpret the following expectations $E[X|X\leq c]$ and $E[X1_{X\leq c}]$? I know its a very simple question but I do not know how to interpret these two expectations and any practical examples will be very helpful. Thanks in advance.
For Example:
If we have a class of students. The students heights are random. Can we have explanation for conditional expectation and the other expectation for this example. Thanks in advance.
On
The condition is supposed to be interpreted as narrowing down the domain of possibilities. Conditional expectation is the recalculated expectation by resetting probabilities to zero for those outcomes not satisfying the given condition, and reassigning the probability for those that do.
Example: When a fair die is thrown the expectation is $\frac16(1+2+3+4+5+6)=7/2$. (all the 6 outcomes have probability 1/6). Now put the condition that the outcome is a square number. So new probabilities are 1/2 for 1 and 4 (the square numbers) and 0 for others.
So the revised (conditional expectation) is $\frac12(1+4)=5/2$
$\mathsf E(X\mid X\leq c)$ is the conditionally expectation for the random variable $X$ when given the event that $X\leq c$.
$\mathsf E(X\,\mathbf 1_{ X\leq c})$ is the expected value of the product of random variable $X$ and the indicator random variable for the event that $X\leq c$.
When the event $X\leq c$ has positive probability mass, then they will be related as: $$\mathsf E(X\,\mathbf 1_{X\leq c})=\mathsf E(X\mid X\leq c)\cdot \mathsf P(X\leq c)$$
If $X$ measures the height of students, then $\mathsf E(X\mid X\leq 1\text{m})$ is the expected height of students who are less than one metre heigh, while $\mathsf E(X~\mathbf 1_{X\leq 1\text{m}})$ is the expected height of students when we record $0$ for all students over one metre.