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We know that covariance can be written as a function of marginals and joint CDFs, namely $$\newcommand{\cov}{\operatorname{cov}}\newcommand{\d}{\mathrm{d}}\cov(X,Y) = \iint (F_{X,Y}(x,y) - F_X(x)F_Y(y))\,\d x\,\d y$$ How can we rewrite $E(Y|X<a)$ in the same way as function of CDFs?

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$$P[Y<y|X<a]=\frac{F_{X,Y}(a,y)}{F_X(a)}$$ So, $$E[Y|X<a]=\int{1-\frac{F_{X,Y}(a,y)}{F_X(a)}dy}=\int{\frac{F_X(a)-F_{X,Y}(a,y)}{F_X(a)}dy}$$

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  • $\begingroup$ We are assuming that $Y$ has positive support here, right? $\endgroup$ – AnonA Jan 29 at 12:20
  • $\begingroup$ Yes, in the first step. $\endgroup$ – John L Jan 29 at 17:27
  • $\begingroup$ In the first step, the conditional distribution of Y given X<a needs to have positive support. If the distribution of Y given X<a has support that is bounded below by -M where M is a positive number, then it can still work with some easy modification since Y+M now has positive support and $E[Y|X<a]=-M+E[Y+M|X<a]$. $\endgroup$ – John L Jan 29 at 18:00

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