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**1. Homework Statement**

I have two I.I.D random variables. I want the conditional expectation of Y given Y is less than some other independent random variable Z.

[tex] E(Y \, \vert \, Y < z) = \dfrac{\int_0^{z} y \cdot f(y) \, dy}{F(z)} [/tex]

Where f(y) is the pdf of Y and F(z) is the cdf for Z

**3. The Attempt at a Solution**

I've searched the book and the web, but all I find is the formula for conditional expectation for [tex] E(X | Y = y) [/tex] for joint distributions and the like. Is my formula correct?