Given E[v|X=x]=g[x] and the pdf of X (f[x]), how to calculate E[v|x>=x0]? The pdf of V or the joint pdf of V,X are unknown. My guess is that this problem has no solution.
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$\begingroup$ Closed. This question isn't appropriate for mathoverflow. Please check the FAQ for suggestions of other sites. $\endgroup$– Kim MorrisonCommented May 16, 2010 at 19:13
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1 Answer
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So if $p(v,x)$ is the unknown pdf,
$f(x) = \int p(v,x) \mathrm{d}v$
$g(x) = E[v|X=x] = \int v \ P[v|X=x] \mathrm{d}v = \int v \ \frac{p(v,x)}{f(x)} \mathrm{d}v$
and then
$E[v|X \ge x_0] = \frac{1}{P[X \ge x_0]} \int_{x_0}^\infty E[v|X = x] P[x] \mathrm{d}x = \frac{ \int_{x_0}^\infty f(x)g(x) \mathrm{d}x}{\int_{x_0}^\infty f(x)\mathrm{d}x}$
