Last updated: 2022-03-01

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# Pre-requisites

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

# 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)

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 [https://en.wikipedia.org/wiki/Stable_distribution]

sessionInfo()
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

locale:
[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):
[1] Rcpp_1.0.7       whisker_0.4      knitr_1.36       magrittr_2.0.2
[5] workflowr_1.7.0  R6_2.5.1         rlang_0.4.12     fastmap_1.1.0
[9] fansi_0.5.0      highr_0.9        stringr_1.4.0    tools_4.1.0
[13] xfun_0.28        utf8_1.2.2       git2r_0.29.0     jquerylib_0.1.4
[17] htmltools_0.5.2  ellipsis_0.3.2   rprojroot_2.0.2  yaml_2.2.1
[21] digest_0.6.28    tibble_3.1.6     lifecycle_1.0.1  crayon_1.4.2
[25] later_1.3.0      vctrs_0.3.8      fs_1.5.0         promises_1.2.0.1
[29] glue_1.5.0       evaluate_0.14    rmarkdown_2.11   stringi_1.7.5
[33] compiler_4.1.0   pillar_1.6.4     httpuv_1.6.3     pkgconfig_2.0.3 

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