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# Linear combinations of normal random variables

One property that makes the normal distribution extremely tractable from an analytical viewpoint is its closure under linear combinations: the linear combination of two independent random variables having a normal distribution also has a normal distribution. The following sections present a multivariate generalization of this elementary property and then discuss some special cases.

## Linear transformation of a multivariate normal random vector

A linear transformation of a multivariate normal random vector also has a multivariate normal distribution, as illustrated by the following:

Proposition Let be a multivariate normal random vector with mean and covariance matrix . Let be an real vector and an full-rank real matrix. Then the random vector defined by:has a multivariate normal distribution with meanand covariance matrix

Proof

This is proved using the formula for the joint moment generating function of the linear transformation of a random vector. The joint moment generating function of is Therefore, the joint moment generating function of is:which is the moment generating function of a multivariate normal distribution with mean and covariance matrix . Note that needs to be positive definite in order to be the covariance matrix of a proper multivariate normal distribution, but this is implied by the assumption that is full-rank. Therefore, has a multivariate normal distribution with mean and covariance matrix , because two random vectors have the same distribution when they have the same joint moment generating function.

The following examples present some important special cases of the above property.

### Example 1 - Sum of two independent normal random variables

The sum of two independent normal random variables has a normal distribution, as stated in the following:

Example Let be a random variable having a normal distribution with mean and variance . Let be a random variable, independent of , having a normal distribution with mean and variance . Then, the random variable defined as:has a normal distribution with mean and variance

Proof

First of all, we need to use the fact that mutually independent normal random variables are jointly normal: the random vector defined ashas a multivariate normal distribution with mean and covariance matrix

We can write:where

Therefore, according to the above proposition on linear transformations, has a normal distribution with mean:and variance:

### Example 2 - Sum of more than two mutually independent normal random variables

The sum of more than two independent normal random variables also has a normal distribution, as shown in the following:

Example Let be mutually independent normal random variables, having means and variances . Then, the random variable defined as:has a normal distribution with mean and variance

Proof

This can be obtained, either generalizing the proof of the proposition in Example 1, or using the proposition in Example 1 recursively (starting from the first two components of , then adding the third one and so on).

### Example 3 - Linear combinations of mutually independent normal random variables

The properties illustrated in the previous two examples can be further generalized to linear combinations of mutually independent normal random variables:

Example Let be mutually independent normal random variables, having means and variances . Let be constants. Then, the random variable defined as:has a normal distribution with mean and variance

Proof

First of all, we need to use the fact that mutually independent normal random variables are jointly normal: the random vector defined ashas a multivariate normal distribution with mean and covariance matrix

We can write:where

Therefore, according to the above proposition on linear transformations, has a (multivariate) normal distribution with mean:and variance:

### Example 4 - Linear transformation of a normal random variable

A special case of the above proposition obtains when has dimension (i.e. it is a random variable):

Example Let be a normal random variable with mean and variance . Let and be two constants (with ). Then the random variable defined by:has a normal distribution with meanand variance

Proof

This is just a special case () of the above proposition on linear transformations.

### Example 5 - Linear combinations of mutually independent normal random vectors

The property illustrated in Example 3 can be generalized to linear combinations of mutually independent normal random vectors.

Example Let be mutually independent normal random vectors, having means and covariance matrices . Let be real full-rank matrices. Then, the random vector defined as:has a normal distribution with mean and covariance matrix

Proof

This is a consequence of the fact that mutually independent normal random vectors are jointly normal: the random vector defined ashas a multivariate normal distribution with mean and covariance matrix

Therefore, we can apply the above proposition on linear transformations to the vector .

## Solved exercises

Below you can find some exercises with explained solutions:

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