# 3.11 Mixtures of Distributions

Rrandom variable has a **mixed distribution** if its value will be obtained by randomly drawing from one of the values to be obtained for two or more other random variables. The random variable’s distribution is a **mixture** of the other random variables’ distributions.

Consider an experiment. You randomly draw two numbers, one from an *N*(0,4) distribution and the other from an *N*(0,9) distribution.10 Next, you flip a fair coin. If it comes up “heads”, you set *X* equal to the number drawn from the *N*(0,4) distribution. Otherwise, you set *X* equal to the number drawn from the *N*(0,9) distribution. The number *X* that will result from this experiment has a **mixed normal distribution** with PDF

[3.126]

This is the weighted average of the PDFs of the two normal distributions. More generally, consider *m* random variables *X _{k}*, each with PDF ϕ

*. Define*

_{k}*m*weights ξ

*> 0 that sum to 1. Then the random variable*

_{k}*X*that has PDF

[3.127]

has a mixed distribution.

###### 3.11.1 Parameters of mixed distributions

Consider a random variable *X* with a mixed distribution as described above. The *X _{k}* have means μ

*and standard deviations σ*

_{k}*. Then*

_{k}*X*has mean μ and standard deviation σ given by

[3.128]

[3.129]

Calculating a *q*-quantile of *X* requires that we solve a nonlinear system of equations, which can be done with Newton’s method. The *q*-quantile is that value *x* such that

[3.130]

so we seek probabilities *q*_{1}, *q*_{2}, … , *q _{m}* such that

[3.131]

while

[3.132]

The desired *q*-quantile *x* of *X* then equals any of—all of—these:

[3.133]

These conditions are motivated for the case *m* = 2 in Exhibit 3.23.

*X*at the bottom is a mixture of normal random variables

*X*

_{1}and

*X*

_{2}. Weights are ξ

_{1}= 0.4 and ξ

_{2}= 0.6. With the distributions stacked one above the other, you can see how the shapes of the upper two determine the shape of the bottom one. To find a

*q*-quantile

*x*of

*X*, we must find probabilities

*q*

_{1}and

*q*

_{2}such that ξ

_{1}

*q*

_{1}+ ξ

_{2}

*q*

_{2}=

*q*while the

*q*

_{1}-quantile

*x*

_{1}of

*X*

_{1}equals the

*q*

_{2}-quantile

*x*

_{2}of

*X*

_{2}.

###### 3.11.2 Mixed-normal distributions

Since a normal distribution is defined by a mean and standard deviation, a mixed-normal distribution *N ^{m}*(

**μ**,

**σ**,

^{2}**ξ**) is defined with a vector

**μ**of means, a vector

**σ**of variances, and a vector

^{2}**ξ**of weights:

[3.134]

where the weights ξ* _{k}* > 0 sum to 1.

Mixed-normal distributions are useful for modeling multimodal or leptokurtic distributions. Exhibit 3.24 illustrates PDFs for two mixed-normal distributions. The first is weighted 0.6 in an *N*(–1,1) distribution and 0.4 in an *N*(2,1) distribution to achieve a bimodal distribution. The second is evenly weighted in *N*(0,1) and *N*(0,9) distributions to achieve a leptokurtic distribution.

*N*(–1,1) distribution and 0.4 in an

*N*(2,1) distribution to achieve a bimodal distribution. The second is evenly weighted in

*N*(0,1) and

*N*(0,9) distributions to achieve a leptokurtic distribution.

###### 3.11.3 Mixed joint-normal distributions

While our discussion of mixed distributions has focused on random variables, similar concepts generalize for random vectors.

Market professionals often observe that market correlations seem exaggerated during large market swings. This phenomenon can be modeled with a mixture of joint-normal distributions—one with low variances and modest correlations and the other with high variances and more extreme correlations.

Consider vectors of *n*-dimensional mean vectors, *n × **n* covariance matrices, and scalar weights:

[3.135]

where the weights ξ* _{k}* > 0 sum to 1. These define a

**mixed joint-normal distribution**with PDF

[3.136]

where ϕ* _{k}*(

**) ~**

*x**N*(

_{n}**μ**

*,*

_{k}**Σ**

*).*

_{k}###### Exercises

Derive formulas [3.128] and [3.129].

Solution

Consider random variable *X* ~ *N*^{ 2}(**μ**,**σ ^{2}**,

**ξ**), where:

[3.137]

Calculate the mean, standard deviation, and .25-quantile of *X*.

Solution