Last updated: 2026-02-10

Checks: 7 0

Knit directory: fiveMinuteStats/analysis/

This reproducible R Markdown analysis was created with workflowr (version 1.7.1). The Checks tab describes the reproducibility checks that were applied when the results were created. The Past versions tab lists the development history.

R Markdown file: up-to-date

Great! Since the R Markdown file has been committed to the Git repository, you know the exact version of the code that produced these results.

Environment: empty

Great job! The global environment was empty. Objects defined in the global environment can affect the analysis in your R Markdown file in unknown ways. For reproduciblity it’s best to always run the code in an empty environment.

Seed: set.seed(12345)

The command set.seed(12345) was run prior to running the code in the R Markdown file. Setting a seed ensures that any results that rely on randomness, e.g. subsampling or permutations, are reproducible.

Session information: recorded

Great job! Recording the operating system, R version, and package versions is critical for reproducibility.

Cache: none

Nice! There were no cached chunks for this analysis, so you can be confident that you successfully produced the results during this run.

File paths: relative

Great job! Using relative paths to the files within your workflowr project makes it easier to run your code on other machines.

Repository version: d1767e4

Great! You are using Git for version control. Tracking code development and connecting the code version to the results is critical for reproducibility.

The results in this page were generated with repository version d1767e4. See the Past versions tab to see a history of the changes made to the R Markdown and HTML files.

Note that you need to be careful to ensure that all relevant files for the analysis have been committed to Git prior to generating the results (you can use wflow_publish or wflow_git_commit). workflowr only checks the R Markdown file, but you know if there are other scripts or data files that it depends on. Below is the status of the Git repository when the results were generated: 


Ignored files:
    Ignored:    analysis/bernoulli_poisson_process_cache/

Untracked files:
    Untracked:  analysis/.#mvnorm.Rmd

Unstaged changes:
    Modified:   analysis/mvnorm.Rmd

Note that any generated files, e.g. HTML, png, CSS, etc., are not included in this status report because it is ok for generated content to have uncommitted changes.

These are the previous versions of the repository in which changes were made to the R Markdown (analysis/mvnorm_00.Rmd) and HTML (docs/mvnorm_00.html) files. If you’ve configured a remote Git repository (see ?wflow_git_remote), click on the hyperlinks in the table below to view the files as they were in that past version.

File Version Author Date Message
Rmd d1767e4 Peter Carbonetto 2026-02-10 wflow_publish("analysis/mvnorm_00.Rmd")
html 3f62e11 Peter Carbonetto 2026-02-10 Ran wflow_publish("analysis/mvnorm_00.Rmd").
Rmd 8fc2bbe Peter Carbonetto 2026-02-10 Added pdf version of mvnorm_00 vignette.
html a32df04 Matthew Stephens 2022-03-01 Build site.
Rmd ea5b4f3 Matthew Stephens 2022-03-01 workflowr::wflow_publish("mvnorm_00.Rmd")
html 77d271e Matthew Stephens 2022-03-01 Build site.
Rmd a9ceaa1 Matthew Stephens 2022-03-01 workflowr::wflow_publish("mvnorm_00.Rmd")

See here for a PDF version of this vignette.

Prerequisites

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 \(X_1 = Z_1 + 0.1 Z_2\), \(X_2 = Z_1 - 0.1 Z_2\). 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 \(X_2\) should also be big, and when \(X_1\) is small \(X_2\) should also be 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
par(mfcol = c(1,3))
hist(X1,breaks = 32)
hist(X2,breaks = 32)
plot(X1,X2,main = "scatterplot of (X1,X2)",ylim = c(-4,4),asp = 1)

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 1,000 samples from a bivariate normal distribution. The prefix “bi” means two, referring to the fact that here we are looking at two variables, \(X_1\) and \(X_2\). The ideas here can be extended to more variables, and the resulting distribution is called the “multivariate normal” (multi = two or more). 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 \(2 \times 2\) 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 \(\Sigma = AA^T\). (Here, \(A^T\) means the transpose of the matrix \(A\).) We write \(X \sim N_2(\mu, \Sigma)\).

Example

We can redo the example above with vector and matrix notation, \(\mu = (0,0)^T\) and \(A = \left[\begin{array}{cc} 1 & 0.1 \\ 1 & -0.1 \end{array}\right]\). Here for clarity we just simulate a single sample instead of 1,000:

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 \(\Sigma = AA^T\):

Sigma <- A %*% t(A)
Sigma
#      [,1] [,2]
# [1,] 1.01 0.99
# [2,] 0.99 1.01

sessionInfo()
# R version 4.3.3 (2024-02-29)
# Platform: aarch64-apple-darwin20 (64-bit)
# Running under: macOS 15.7.1
# 
# Matrix products: default
# BLAS:   /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRblas.0.dylib 
# LAPACK: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.11.0
# 
# locale:
# [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
# 
# time zone: America/Chicago
# tzcode source: internal
# 
# attached base packages:
# [1] stats     graphics  grDevices utils     datasets  methods   base     
# 
# loaded via a namespace (and not attached):
#  [1] vctrs_0.6.5       cli_3.6.5         knitr_1.50        rlang_1.1.6      
#  [5] xfun_0.52         stringi_1.8.7     promises_1.3.3    jsonlite_2.0.0   
#  [9] workflowr_1.7.1   glue_1.8.0        rprojroot_2.0.4   git2r_0.33.0     
# [13] htmltools_0.5.8.1 httpuv_1.6.14     sass_0.4.10       rmarkdown_2.29   
# [17] evaluate_1.0.4    jquerylib_0.1.4   tibble_3.3.0      fastmap_1.2.0    
# [21] yaml_2.3.10       lifecycle_1.0.4   whisker_0.4.1     stringr_1.5.1    
# [25] compiler_4.3.3    fs_1.6.6          Rcpp_1.1.0        pkgconfig_2.0.3  
# [29] later_1.4.2       digest_0.6.37     R6_2.6.1          pillar_1.11.0    
# [33] magrittr_2.0.3    bslib_0.9.0       tools_4.3.3       cachem_1.1.0