# Difference between the terms 'joint distribution' and 'multivariate distribution'?

I am writing about using a 'joint probability distribution' for an audience that would be more likely to understand 'multivariate distribution' so I am considering using the later. However, I do not want to loose meaning while doing this.

Wikipedia seems to indicate that these are synonyms.

Are they? If not, why not?

The terms are basically synonyms, but the usages are slightly different. Think about the univariate case: you may talk about "distributions" in general, you might more specifically refer to "univariate distributions", and you refer to "the distribution of $X$". You don't normally say "the univariate distribution of $X$".

Similarly, in the multivariate case you may talk about "distributions" in general, you might more specifically refer to "multivariate distribution", and you refer to "the distribution of $(X,Y)$" or "the joint distribution of $X$ and $Y$". Thus the joint distribution of $X$ and $Y$ is a multivariate distribution, but you don't normally say "the multivariate distribution of $(X,Y)$" or "the multivariate distribution of $X$ and $Y$".

• +1. On Google: "the univariate distribution of" has 25,600 hits. "the joint distribution of": 1,080,000. "the multivariate distribution of": 85,100. "the bivariate distribution of": 89,800. This sounds like the "joint" version is popular with "univariate," "bivariate," and "multivariate" occasionally used, each with similar frequencies. These likely are used in circumstances requiring clarification. (I have often seen "the univariate distribution of" used in this sense.)
– whuber
Feb 15, 2011 at 23:30

I'd be inclined to say that "multivariate" describes the random variable, i.e., it is a vector, and that the components of a multivariate random variable have a joint distribution. "Multivariate random variable" sounds a bit strange, though; I'd call it a random vector.

• I've bumped into a case where the two variables are of different shapes (e.g. ones a scalar, the other a matrix). In this case, can it be a random vector?
– joel
Apr 7, 2021 at 14:31

The canonical textbooks describing properties of the various probability distributions by Johnson & Kotz and later co-authors are entitled Univariate Discrete Distributions, Continuous Univariate Distributions, Continuous Multivariate Distributions and Discrete Multivariate Distributions. So I think you're on safe ground describing a distribution as 'multivariate' rather than 'joint'.

Conflict of interest statement: The author is a member of Wikipedia:WikiProject Statistics.

I think they are mostly synonyms, and that if there is any difference, it lies in details that are likely irrelevant to your audience.

I would be careful to say a joint distribution is synonymous with a multivariate distribution. For example a joint normal distribution can be a multivariate normal distribution or a product of univariate normal distributions.

A univariate normal distribution has a scalar mean and a scalar variance, so for the univariate (one dimensional) random variable $x$ distributed according to a normal we have $p(x) = N(x ; \mu, \sigma)$.

A multivariate normal distribution has mean vector of length $n>1$ and a covariance matrix of size $n \times n$. For two univariate random variables $x,y$ they can be jointly distributed according to a multivariate normal distribution $p(x,y) = \mathcal{N}([x \ y]^\intercal ; [\mu_x \ \mu_y]^\intercal , \Sigma_{xy})$.

However, if the covariance matrix of the multivariate distribution is a diagonal matrix, this means that x and y have zero correlation (are independent) and so the joint distribution can be a product of univariate Gaussians, $p(x,y) = N(x ; \mu_x, \sigma_x)*N(y ; \mu_y, \sigma_y)$.

Therefore the joint distribution is not really synonymous with the multivariate in the case of independent variables.

https://en.wikipedia.org/wiki/Joint_probability_distribution#Joint_distribution_for_independent_variables

• I disagree with this answer. A joint normal distribution is a specific form that is also called a multivariate normal distribution of which the product of univariate normal distributions is a special case, not something to be called out separately. All multivariate distributions of finite-variance random variables, whether multivariate normal or not, possess mean vectors and covariance matrices. Finally, normal random variables don't have to have a multivariate normal distribution: see this answer for a wealth of examples . Dec 1, 2017 at 16:40