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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.

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$\mathsf E(X\mid X\leq c)$ 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.

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  • $\begingroup$ Thank you so much for your answer. But what will be the interpretation. For example if we have a class of students with random heights. If $X$ represent the height of student. Then how to interpret $E[X1_{X\leq 5 feet}]$ and $E[X|X\leq 5 feet]$? I mean if we want to find these expectations then in which way the procedure will be different for finding both of them? $\endgroup$ Commented Mar 28, 2018 at 5:45
  • $\begingroup$ Thank you for your example. We say $E(X|X \leq c)$ as the conditional expectation. What is the proper name of $E(X1_{X\leq c})$? Is it also called conditional expectation? $\endgroup$ Commented Mar 28, 2018 at 6:21
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    $\begingroup$ Actually, I think there is a word for it that's used in probability theory or statistics. Something to do with cut off or restriction. Might have something to do with stopping times. Not sure. Based on Graham Kemp's explanation, I thought of options in finance. I think it may be helpful to consider $X1_{X \le c} + c1_{X > c}$. For $c=0$ we have $X1_{X \le c} + c1_{X > c} = X1_{X \le 0}$. For $c=1$, we have $X1_{X \le 1} + 1_{X > 1} = \min\{X,c\}$. Thus, $X1_{X \le c} = \min\{X,c\} - c1_{X>c} \to E[X1_{X \le c}] = E[\min\{X,c\}] - cP(X>c)$. $\endgroup$ Commented Apr 6, 2018 at 13:55
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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 $\frac12(1+4)=5/2$

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  • $\begingroup$ Thank you for your answer. Here is what I understood. We say $5/2$ is the answer when we know that the outcome was a square number. Right? and the other expectation can we write. The expectation of the dice outcome for square numbers. Is it right? $\endgroup$ Commented Mar 28, 2018 at 6:13
  • $\begingroup$ Going back to your example of heights of students. Assume the class has students of various races. Now the conditional expectation of height given the condition that student is of particular race could be different. $\endgroup$ Commented Mar 28, 2018 at 6:32

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