Last updated: 2022-03-01

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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 \(var(X)= a^2 var(Z_1) + b^2 var(Z_2)\).


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).



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 at []

R version 4.1.0 Patched (2021-07-20 r80657)
Platform: aarch64-apple-darwin20 (64-bit)
Running under: macOS Monterey 12.2

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.1-arm64/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.1-arm64/Resources/lib/libRlapack.dylib

[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

loaded via a namespace (and not attached):
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[13] xfun_0.28        utf8_1.2.2       git2r_0.29.0     jquerylib_0.1.4 
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