Name for the expression E(Y|X)*Prob(X)

#1
I am working with a term which is an expectation conditional on an event, multiplied by the probability of the event occuring. For example, the term might be int_{-inf}^0 x f(x). Obviously this is not an expectation since int_{-inf}^0 f(x) < 1, but I need to give it a name. I'm hoping that it has a name in the literature, but I haven't been able to find one anywhere. If there is no standard name, I'm going to have to invent one. Can I call it a "probability weighted average" or does "average" necessarily imply that the weights integrate to one. If so, then what about a "probability weighted value"?
 

BGM

TS Contributor
#2
I remember that I should have read a name about this before which I think is a good
description for this, but I forget the exact name now.

A term which is quite close (but not exactly the same) to your description is "partial expectation".
 
#3
If \(Y:\Omega \rightarrow \mathbb{R}\) and \(X\) is some event, then \(E[Y|X] = \frac{E[Y\mathbb{I}_X]}{P(X)}\). So your expression is just \(E[Y\mathbb{I}_X]\), where \(\mathbb{I}_X\) is 1 when \(\omega \in X\) and is 0 otherwise.
 
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Dragan

Super Moderator
#4
Can I call it a "probability weighted average" or does "average" necessarily imply that the weights integrate to one. If so, then what about a "probability weighted value"?
That's likely not a good idea because it's confusing with the term "probability weight moments" where the integral also includes the distribution function (raised to powers) as well as the density function and your still integrating over the entire region....i.e.

\(\beta _{r}=\int_{-\infty }^{\infty }x\left ( F\left ( x \right ) \right )^{r}f\left ( x \right )dx\)
 

Dragan

Super Moderator
#6
Thanks Dragan, could it be the "0th probability weight moment"?
Well, I think you're implying r=0 in the equation I wrote. That would be the usual Mean (Expected value of X)...r=1 yields one-half of the Coefficient of Mean Difference (or Gini's index of spread) when you take 2*Beta_1 - Beta_0.