# 3.13.5 Other Parameters of Quadratic Polynomials of Multivariate Normal Random Vectors

###### 3.13.5  Other Parameters

We can calculate any central moment of Y. This is simply a matter of multiplying out the formula for the desired central moment and substituting in values for moments. Consider the third central moment:

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where μ = E(Y). The variance of Y is, by our result from Exercise 3.15,

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The skewness η1 and kurtosis η2 are obtained as

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Quantiles of Y can be approximated using the Cornish-Fisher (1937) expansion, which we discuss in the next section. They can be calculated exactly using the inversion theorem that we discuss in Section 3.16.

###### Exercises
3.42

Consider random vector X ~ N3(μ,Σ) with

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Let

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where

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Express Y as a linear polynomial of independent chi-squared and normal random variables.

3.43

Calculate the mean and standard deviation of the random variable Y of the previous exercise.