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Assume $(X,Y) \sim \mathcal{N}_{2}(\mu, \Sigma)$ is a bivariate normal random vector. Let $Z = \max(0,X)$. I am now looking for the moments of $Z$ given that $Y<0$, in particular for $$\mathbb{E}[Z|Y<0] \text{ and } \mathbb{V}[Z|Y<0]$$

I know that $(X|Y=y) \sim \mathcal{N}\Big(\mu_1 + \rho\frac{\sigma_1}{\sigma_2}(y-\mu_2), (1- \rho^2)\sigma_1^2\Big)$, but how to proceed from this?

kjetil b halvorsen
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Claudio Moneo
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  • https://stats.stackexchange.com/questions/356023/ gives the expectation and presents methods to compute the variance. – whuber Jan 29 '22 at 15:39
  • @whuber I'll have a look, but probably some adjustments have to be made since $Z$ is not Gaussian – Claudio Moneo Jan 29 '22 at 15:49
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    Good point: it is a mixture of a truncated Gaussian and a constant. Mixture moments are easy to compute from those of their components. – whuber Jan 29 '22 at 16:12
  • @whuber Maybe I just don't see it, but it leads to a bivariate normal distribution that is truncated in both variables. How can I get the expectation of that? – Claudio Moneo Jan 30 '22 at 15:15

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