Expectation of truncated distribution

gadelke

New member
Joined
Jun 1, 2018
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10
Suppose that
gif.latex
, where
gif.latex
is an arbitrary continuous PDF with CDF
gif.latex
, and
gif.latex
, where X is a random variable and x is a realisation.

What is the following expectation:
gif.latex
?

i.e. given that I know some random variable is less than some constant, what is the expectation of that random variable?

I have tried to follow the methods used by:
https://en.wikipedia.org/wiki/Truncated_distribution#Expectation_of_truncated_random_variable
https://math.stackexchange.com/questions/89459/what-is-ex-mid-xc-in-terms-of-pxc
https://www.quora.com/What-is-the-expected-value-of-Y-given-that-Y-is-greater-than-C

However, I do not understand what I is in the stackexhange response, and I do not understand what
gif.latex
refers to in the quora answer. My understanding is that
gif.latex
for the denominator of that question?

In any case, I'm not completely sure how to formulate that expectation. Any help would be greatly appreciated. Thank you!
 
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