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I have 4 r.v $(X,Y,W,Z)$ distributed as a multivariate normal.

Mean and variance-covariance matrix are known.

Is it possible to calculate, for example $\mathrm{Prob} (aX+bY < k, W >0,Z >0, W-Z >0)$, where $k$ is a scalar?

If not, which additional information do I need?

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In general multivariate normal probabilities are non-trivial and not usually solvable algebraically. However, on your specific question we can get somewhere.

Note that if $(X,Y,W,Z)$ is multivariate normal then $(k-aX+bY,W,Z,W-Z)$ will also be mutivariate normal (if degenerate in this case), so the problem reduces to one of finding whether a multivariate normal vector is all positive.

  1. If the means of the new set of variates are zero this is an example of finding an orthant probability, a classical problem on which some results are available.

    The book by Genz and Bretz (2009), Computation of Multivariate Normal and t Probabilities lists explicit formulas for up to trivariate normal,

    $P_2= \frac14+\frac{1}{2\pi}\sin^{-1}(\rho_{12})$

    $P_3= \frac18+\frac{1}{4\pi}\{\sin^{-1}(\rho_{12})+\sin^{-1}(\rho_{23})+\sin^{-1}(\rho_{13}) \}$

    and for even dimension gives a dimension-halving formula. As it happens, the relevant formulas are in the sample pages available on the book's web-page

    But in your specific, case, we can in fact simplify further and directly apply the trivariate formula there (see the next section).

  2. In the more general case of nonzero means. Note that if $Z>0$ and $W-Z>0$, then $W>0$ is automatic. Consequently, let our vector be $(k-aX+bY,Z,W-Z)$, compute its mean and covariance matrix; let $(T,U,V)$ be mean-corrected versions of the above vector. This is discussed in the trivariate section of the book that's also in the above linked extract.

  3. The paper by Genz 1992 ("Revised Computation of Multivariate normal probabilities," J.Comp. Graph. Stat, 1, pp. 141-149; working paper here) discusses a number of approximations for the general case.

  4. The R package mvtnorm includes algorithms for calculation of multivariate normal probabilities.

For high dimensional probabilities if I didn't have premade routines to call, I'd probably be leaning toward using various Monte Carlo methods (which are widely used for these problems).

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