Maybe I have not stated the question clearly.

A positive random variable X varies in the range [a,b] with pdf f_X(x).
Suppose a real number r greater than a and less than b, i.e. a<r<b. A value
of the random variable will fall in the range [a,r] or the range [r,b].
Define the event

A = value of random variable falls in the range [a,r];
B = value of random variable falls in the range [r,b];

Then, how to compute the expected value of X with the value in the range
[a,r] and in the range [r,b]. namely, to find

E(X|A) and E(X|B)

Thanks in advance!


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