There is three-variable multivariate normal distribution. Denote 3 variables with $X_1$, $X_2$, $X_3$. Let $\mu_i$ be means, and $\sigma_i^2$ variances of respective variables, and let $\Sigma$ be covariance matrix. Can someone help me with deriving expressions for:

$$E(X_1|X_2>a,X_3>b)$$ and $${\rm Var}(X_1|X_2>a, X_3>b)$$

Where $a$ and $b$ are some real numbers.

  • 2
    $\begingroup$ The trivariate case is messy. See the paper by: Tallis (1961), The moment generating function of the truncated multinormal distribution. Journal of the Royal Statistical Society, Series B, 23(1):223–229 (which includes examples) ... or see some of the recent work by Wilhelm and Manjunath such as: journal.r-project.org/archive/2010-1/… $\endgroup$
    – wolfies
    Jan 26, 2015 at 17:41


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