Last time we took a look at how we could define multivariate normally distributed random variables with linear functions of multiple independent standard univariate normal random variables.

Specifically, given a

has linearly dependent normally distributed elements, a mean vector of

where

We got as far as deducing the characteristic function and the probability density function of the multivariate normal distribution, leaving its cumulative distribution function and its complement aside until we'd implemented both them and the random variable itself, which we shall do in this post.

Specifically, given a

**Z**whose elements are independent standard univariate normal random variables, a constant vector**μ**and a constant matrix**L****Z**' =

**L**×

**Z**+

**μ**

has linearly dependent normally distributed elements, a mean vector of

**μ**and a covariance matrix of**Σ**' =

**L**×

**L**

^{T}

where

**L**^{T}is the transpose of**L**in which the rows and columns are switched.We got as far as deducing the characteristic function and the probability density function of the multivariate normal distribution, leaving its cumulative distribution function and its complement aside until we'd implemented both them and the random variable itself, which we shall do in this post.

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