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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)

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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.]

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  • $\begingroup$ 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

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