# Simulate Bivariate & Multivariate Normal Distribution in R (2 Examples)

This post explains how to **draw a random bivariate and multivariate normal distribution** in the R programming language.

Table of contents:

- Example 1: Bivariate Normal Distribution in R
- Example 2: Multivariate Normal Distribution in R
- Video, Further Resources & Summary

Let’s do this.

## Example 1: Bivariate Normal Distribution in R

Example 1 explains how to generate a random bivariate normal distribution in R.

First, we have to install and load the MASS package to R:

install.packages("MASS") # Install MASS package library("MASS") # Load MASS package |

install.packages("MASS") # Install MASS package library("MASS") # Load MASS package

In case we want to create a reproducible set of random numbers, we also have to set a seed:

set.seed(98989) # Set seed for reproducibility |

set.seed(98989) # Set seed for reproducibility

Then, we have to specify the data setting that we want to create. The following R code specifies the sample size of random numbers that we want to draw (i.e. 1000), the means of our two normal distributions (i.e. 5 and 2), and the variance-covariance matrix of our two variables:

my_n1 <- 1000 # Specify sample size my_mu1 <- c(5, 2) # Specify the means of the variables my_Sigma1 <- matrix(c(10, 5, 3, 7), # Specify the covariance matrix of the variables ncol = 2) |

my_n1 <- 1000 # Specify sample size my_mu1 <- c(5, 2) # Specify the means of the variables my_Sigma1 <- matrix(c(10, 5, 3, 7), # Specify the covariance matrix of the variables ncol = 2)

After specifying all our input arguments, we can apply the mvrnorm function of the MASS package as follows:

mvrnorm(n = my_n1, mu = my_mu1, Sigma = my_Sigma1) # Random sample from bivariate normal distribution |

mvrnorm(n = my_n1, mu = my_mu1, Sigma = my_Sigma1) # Random sample from bivariate normal distribution

**Figure 1: Bivariate Random Numbers with Normal Distribution.**

Figure 1 illustrates the RStudio output of our previous R syntax. The R code returned a matrix with two columns, whereby each of these columns represents one of the normal distributions.

## Example 2: Multivariate Normal Distribution in R

In Example 2, we will extend the R code of Example 1 in order to create a multivariate normal distribution with three variables. As in Example 1, we need to specify the input arguments for the mvrnorm function. However, this time we are specifying three means and a variance-covariance matrix with three columns:

my_n2 <- 1000 # Specify sample size my_mu2 <- c(5, 2, 8) # Specify the means of the variables my_Sigma2 <- matrix(c(10, 5, 2, 3, 7, 1, 1, 8, 3), # Specify the covariance matrix of the variables ncol = 3) |

my_n2 <- 1000 # Specify sample size my_mu2 <- c(5, 2, 8) # Specify the means of the variables my_Sigma2 <- matrix(c(10, 5, 2, 3, 7, 1, 1, 8, 3), # Specify the covariance matrix of the variables ncol = 3)

We can now apply the mvrnorm as we already did in Example 1:

mvrnorm(n = my_n2, mu = my_mu2, Sigma = my_Sigma2) # Random sample from bivariate normal distribution |

mvrnorm(n = my_n2, mu = my_mu2, Sigma = my_Sigma2) # Random sample from bivariate normal distribution

**Figure 2: Multivariate Random Numbers with Normal Distribution.**

Figure 2 illustrates the output of the R code of Example 2. This time, R returned a matrix consisting of three columns, whereby each of the three columns represents one normally distributed variable.

## Video, Further Resources & Summary

Do you need further information on the contents of this article? Then you could have a look at the following video that I have published on my YouTube channel. In the video, I explain the topics of this tutorial:

*The YouTube video will be added soon.*

You could also have a look at the other tutorials on probability distributions and the simulation of random numbers in R:

- Bernoulli Distribution in R
- Beta Distribution in R
- Binomial Distribution in R
- Bivariate & Multivariate Distributions in R
- Cauchy Distribution in R
- Chi-Squred Distribution in R
- Exponential Distribution in R
- F Distribution in R
- Gamma Distribution in R
- Geometric Distribution in R
- Hypergeometric Distribution in R
- Log Normal Distribution in R
- Logistic Distribution in R
- Negative Binomial Distribution in R
- Normal Distribution in R
- Poisson Distribution in R
- Student t Distribution in R
- Studentized Range Distribution in R
- Uniform Distribution in R
- Weibull Distribution in R
- Wilcoxon Signedank Statistic Distribution in R
- Wilcoxonank Sum Statistic Distribution in R

Besides that, you may read some of the other tutorials that I have published on my website:

Summary: In this R programming tutorial you learned how to **simulate bivariate and multivariate normally distributed probability distributions**. In case you have any additional questions, please tell me about it in the comments section below.

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