I'm currently in an Introductory Probability course and am trying to understand probability at a deeper level.

Through some reading, I believe I am correct in saying that the expected value of a probability distribution $X$, $E[X]$, is a functional.

Is the same true of the probability of an event $A$, $P(A)$?

(this question is in part motivated by discussion about whether parentheses or brackets make more sense for the $E$ and $P$ notations, so I'd be interested in any opinions there as well)


Let $A$ is an event, and $\mathbf 1_A$ is its indicator random variable, such that: $$\mathbf 1_A=\begin{cases}1&:& A\\0&:&\text{otherwise}\end{cases}$$

Then from the definition of expectation: $\mathsf E(\mathbf 1_A) ~{= 0\mathsf P(\mathbf 1_A=0)+1\mathsf P(\mathbf 1_A=1)\\ = \mathsf P(A)}$

So if expectation of the random variable is functional, then the probability mass of an event will also be functional.

[Though, noteably, expectation is a linear functional, while probability measure is not linear.]

  • $\begingroup$ Ah, of course. I've seen that exact relationship between $E[x]$ and $P[x]$ before, so that makes total sense. $\endgroup$ – Jack Gallagher Apr 6 '18 at 3:57

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.