We have previously seen how we can generalise normally distributed random variables to multiple dimensions by defining vectors with elements that are linear functions of independent standard normally distributed random variables, having means of zero and standard deviations of one, with

where

So far we have derived and implemented the probability density function and the characteristic function of the multivariate normal distribution that governs such random vectors but have yet to do the same for its cumulative distribution function since it's a rather more difficult task and thus requires a dedicated treatment, which we shall have in this post.

**Z**' =

**L**×

**Z**+

**μ**

where

**L**is a constant matrix,**Z**is a vector whose elements are the independent standard normally distributed random variables and**μ**is a constant vector.So far we have derived and implemented the probability density function and the characteristic function of the multivariate normal distribution that governs such random vectors but have yet to do the same for its cumulative distribution function since it's a rather more difficult task and thus requires a dedicated treatment, which we shall have in this post.

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