How do we show conditional expectation of X given X > c?

[eastla_student]

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?

[Colymbosathon ecplecticos]

E[X] = E[X | X <= c] * P(X <= c) + E[X | X > c] * P(X > c)

Works for any event A and its complement.

[Vorian Atreides]

First, work it out for the discrete case.

Then look at how to convert it to the continuous case.

[eastla_student]

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]