Expected value given that distribution is positive vs. conditional expectation

Question

Referring to Expected value of normal distribution given that distribution is positive

Where is the difference between $E(X$1$_A)$, where $A=[X>0]$, and $E(X∣A)$?

Both seem to express the expected value of $X$ given that $X>0$ which is equal to the conditional expectation

score 2 · Answer 1 · answered Mar 29 '14 at 17:00

2

$$E(X\,\mathbf 1_A)=E(X\mid A)\,P(A)$$ $$E(X\,\mathbf 1_{X\gt0})=\int_0^\infty x\,f_X(x)\,\mathrm dx$$ $$P(X\gt0)=\int_0^\infty f_X(x)\,\mathrm dx$$

answered Mar 29 '14 at 17:00

Did

279,727

Also here trivially P(A)=P(X>0) – James LT Nov 22 '17 at 01:38
@Did We refer to $E(X|A)$ as a "conditional expectation". Does $E(X,\mathbf{1}_{A})$ also have a name? – Aaron Hendrickson Jun 07 '18 at 15:53
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@AaronHendrickson Not an established one. – Did Jun 07 '18 at 16:36

score 0 · Answer 2 · answered Mar 29 '14 at 02:03

0

One is $$ \int_0^\infty x f(x) dx $$ and the other is $$ \frac{\int_0^\infty x f(x) dx}{\int_0^\infty f(x) dx} $$ where $f$ is the pdf for the Normal distribution.

answered Mar 29 '14 at 02:03

Rookatu

1,752

Expected value given that distribution is positive vs. conditional expectation

2 Answers2