Last updated: 2022-03-01

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# Pre-requisites

You need to have basic familiarity with univariate normal distribution, and understand the basic property that linear combinations of normals are also normal.

# Motivating example

Suppose that $$Z_1,Z_2$$ are independent standard normal $$N(0,1)$$ and define $$X_1=Z_1+0.1 Z_2$$ and $$X_2=Z_1-0.1 Z_2$$. What is the joint distribution of $$X_1,X_2$$?

We know from the basic property that $$X_1$$ will be univariate normal, and that $$X_2$$ will be univariate normal. However, they will not necessarily be independent because $$Z_1$$ and $$Z_2$$ were used to compute both. Indeed, you can see that $$X_1$$ and $$X_2$$ might both be expected to be close to $$Z_1$$ (because the 0.1 multiplier on $$Z_2$$ is “relatively small”). So when $$X_1$$ is big we should expect $$X_2$$ will likely be big, and when $$X_1$$ is small we should expect $$X_2$$ will likely small.

The following code illustrates this: the histograms illustrate both $$X_1$$ and $$X_2$$ are normal, and the scatterplot of $$X_1$$ and $$X_2$$ shows they are correlated (and the sample correlation is approximately 0.98).

Z1 = rnorm(1000)
Z2 = rnorm(1000)

X1 = Z1+0.1*Z2
X2 = Z1-0.1*Z2

hist(X1)

Version Author Date
77d271e Matthew Stephens 2022-03-01
hist(X2)

Version Author Date
77d271e Matthew Stephens 2022-03-01
plot(X1,X2, main="scatterplot of (X1,X2)", ylim=c(-4,4), asp=1) #asp=1 sets the scales of X1 and X2 the same

Version Author Date
77d271e Matthew Stephens 2022-03-01
cor(X1,X2)
[1] 0.9798486

# The bivariate normal distribution

In fact the answer to the question “what is the joint distribution of $$X_1,X_2$$” is they have a “bivariate normal distribution”. Thus the scatterplot shown above shows a scatterplot of 1000 samples from a bivariate normal distribution. The prefix “bi” means 2, referring to the fact that here we are looking at 2 variables, $$X_1$$ and $$X_2$$. The ideas here can be extended to more variables, and the resulting distribution is called the “multivariate normal”. The bivariate normal is a special case of the multivariate normal.

## Mean and Covariance Matrix

The bivariate normal distribution has 5 parameters: two means (for $$X_1$$ and $$X_2$$), two variances (for $$X_1$$ and $$X_2$$) and the covariance between $$X_1$$ and $$X_2$$. It is usual to write the mean parameter as a vector $$\mu$$ and the variance and covariance parameters as a 2x2 symmetric matrix $$\Sigma$$, where the diagonal elements of $$\Sigma$$ contain the variances and the off-diagonal elements contain the covariance. $$\Sigma$$ is called the “covariance matrix” (or sometimes the “variance-covariance matrix”).

## General Construction

Suppose $$Z_1,Z_2$$ are independent random variables each with a standard normal distribution $$N(0,1)$$. Let $$Z$$ denote the vector $$(Z_1,Z_2)$$, let $$A$$ be any $$2 \times 2$$ matrix, and $$\mu$$ be any $$r$$-vector. Then the vector $$X = AZ+\mu$$ has a bivariate normal distribution with mean $$\mu$$ and variance-covariance matrix $$\Sigma=AA'$$. (Here $$A'$$ means the transpose of the matrix $$A$$.) We write $$X \sim N_2(\mu,\Sigma)$$.

# Example

We can redo the example above in vector and matrix notation, with $$\mu=(0,0)$$ and $$A=(1,0.1),(1,-0.1)$$. Here for clarity we just simulate a single sample instead of 1000:

mu = c(0,0)
A = rbind(c(1,0.1),c(1,-0.1))
A
     [,1] [,2]
[1,]    1  0.1
[2,]    1 -0.1
z = rnorm(2)
x = mu + A %*% z
x
           [,1]
[1,] -0.5002188
[2,] -0.7154634

It should be clear from the above that in our example the mean is $$\mu=(0,0)$$. What is the covariance matrix $$\Sigma$$? We can compute it from the formula $$\Sigma = AA'$$:

Sigma = A %*% t(A)
Sigma
     [,1] [,2]
[1,] 1.01 0.99
[2,] 0.99 1.01

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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