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Old 07-01-2018, 11:29 PM
eastla_student eastla_student is offline
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Default How do we show conditional expectation of X given X > c?

Hi, I'm having trouble convincing myself that

E( X | X > c) = integral from c to infinity on x * f(x) all divided by P(X>c)

through a chain of equalities. In particular, getting P(X>c) into the equality bothers me. Intuitively, it makes sense, but how do I show it in a chain of equalities?
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Old 07-01-2018, 11:48 PM
Colymbosathon ecplecticos's Avatar
Colymbosathon ecplecticos Colymbosathon ecplecticos is offline
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E[X] = E[X | X <= c] * P(X <= c) + E[X | X > c] * P(X > c)

Works for any event A and its complement.
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Old 07-01-2018, 11:49 PM
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First, work it out for the discrete case.

Then look at how to convert it to the continuous case.
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Old 07-02-2018, 12:10 AM
eastla_student eastla_student is offline
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I see now, if we only care about E(X) over c to infinity then

integral from c to infinity of x*f(x) = E(X) over c to infinity
E[X | X <= c] * P(X <= c) + E[X | X > c] * P(X > c) =
E[X | X <= c] * 0 + E[X | X > c] * P(X > c) =
E[X | X > c] * P(X > c)

so then (integral from c to infinity of x*f(x)) / P(X > c) = E[X | X > c]
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