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#1




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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#2




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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#3




First, work it out for the discrete case.
Then look at how to convert it to the continuous case.
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#4




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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