Good morning,
How can I compute $\mathbb E[X^2|X>Y] $ given X and Y as continuous random variable. $X$ and $Y$ follows the same distribution, in fact $Y=F^{(-1)}(c)$ where c is a given number.
I know that: $\mathbb E[X|X>Y]=\frac{1}{P(X>Y)}\int_{Y}^{+\infty} x\cdot f(x) \cdot dx =\frac{1}{P(X>Y)}\int_{F(Y)}^{F(+\infty)} F^{-1}(t) \cdot dt$
Given the function $F(x)=\int_{-\infty}^x f(u)du$.
However I do not know how I could compute $\mathbb E[X^2|X>Y] $ . Thank you very much for your help.
[This answer is in response to OP's comment above].
If $X$ and $Y$ are independent with densities $f$ and $g$ then $E(X^{2}|X>Y)=\frac {\int_{-\infty} ^{\infty} \int_{y} ^{\infty} x^{2}f(x)g(y)dxdy} {\int_{-\infty} ^{\infty}\int_{y} ^{\infty} f(x)g(y)dxdy} $