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I have two jointly normally distributed variables $s_1$ and $s_2$. I am now searching for the conditional expectation $$ E(s_1|s_1>r_1,\ s_2>r_2) $$ where $r_1$ and $r_2$ are constants. An idea of how to arrive at this and sources where it can be read up would also be greatly appreciated.

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  • $\begingroup$ What do you know about the joint distribution of $s_1$ and $s_2$? This information is essential for determining the conditional expectation. $\endgroup$
    – whuber
    Commented Jun 12, 2014 at 14:21
  • $\begingroup$ $s_1$ and $s_2$ are generally correlated, but assume they have unconditional mean zero and variance one. If they weren't correlated, it'd be easy I think... $\endgroup$
    – Mike
    Commented Jun 12, 2014 at 14:35
  • $\begingroup$ Is the joint distribution bivariate Normal? $\endgroup$
    – whuber
    Commented Jun 12, 2014 at 14:37
  • $\begingroup$ Yes, let's say: $$s_1,s_2 \sim \mathcal{N}(0,1) $$ and the correlation is $\rho$. Sorry, I don't manage to type them together in one bivariate normal... $\endgroup$
    – Mike
    Commented Jun 12, 2014 at 14:48
  • $\begingroup$ Unfortunately, both integrals in the ratio that answers this question are intractable when $\rho\ne 0$ and $\rho \ne \pm 1$ and have to be computed numerically. $\endgroup$
    – whuber
    Commented Jun 12, 2014 at 16:42

1 Answer 1

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To summarize the comments: Since we assume that $s_1$ and $s_2$ jointly follow a standard bivariate normal distribution, with correlation coefficient $\rho$, then the joint density is

$$f(s_1,s_2) = \frac{1}{2 \pi \sqrt{1-\rho^2}} \exp\left\{-\frac{s_1^2 +s_2^2 -2\rho s_1s_2}{2(1-\rho^2)}\right\} $$

We also have

$$E(s_1|s_1>r_1,\ s_2>r_2) = \frac {E(s_1;\{s_1>r_1,\ s_2>r_2\})}{P(s_1>r_1,\ s_2>r_2)}$$

$$=\frac {\int_{r_2}^{\infty}\int_{r_1}^{\infty}s_1f(s_1,s_2)ds_1ds_2}{\int_{r_2}^{\infty}\int_{r_1}^{\infty}f(s_1,s_2)ds_1ds_2} $$

The fact that the conditioning statement includes, and places bounds on, both variables, does not permit us to simplify this ratio of integrals (as would be in the cases described by the OP in the comments). Moreover, as @whuber writes, these integrals do not have an analytical solution for $\rho \ne \{0,\pm1\}$, and must be computed numerically for each $\{r_1, r_2\}$.

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