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See here for a PDF version of this vignette.

Prerequisites

Basic familiarity with the univariate normal distribution.

A statement of the basic property

The simple goal of this vignette is to introduce a basic property of the (univariate) normal distribution: that linear combinations of independent normal variables are also normal.

Formally, suppose \(Z_1\) and \(Z_2\) represent independent, normally distributed random variables. Then for any scalars \(a\) and \(b\), the linear combination \[ X := aZ_1 + bZ_2 \] is also (univariate) normal.

Also, by basic properties of expectation and variance, \(E(X) = aE(Z_1) + bE(Z_2)\) and \(\mathrm{Var}(X) = a^2 \mathrm{Var}(Z_1) + b^2 \mathrm{Var}(Z_2)\).

Example

The following code provides a visual illustration of this idea with \(a = 2\) and \(b = 3\), but it holds for any \(a\) and \(b\).

First we sample some values of \(X\) by randomly generating \(Z_1\) and \(Z_2\), and computing \(X = aZ_1 + bZ_2\):

Z1 <- rnorm(1000)
Z2 <- rnorm(1000)
a <- 2
b <- 3
X <- a*Z1 + b*Z2

The property says that the samples of \(X\) look normal. A quick histogram and qqplot suggest it does. (Of course, this is not a proof that the property holds; it is just an illustration of the idea.)

hist(X)

qqnorm(X)

Addendum: Stable distributions

If you are curious by nature, you might now ask: is the normal distribution the only distribution that satisfies this property? The answer is “no”. For example, \(t\) distributions also satisfy this property. Distributions that satisfy this property are called “stable” distributions. You can read more here.


sessionInfo()
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