Perbedaan antara istilah 'distribusi bersama' dan 'distribusi multivariat'?

15

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?

David LeBauer
sumber

Jawaban:

11

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".

Mark Meckes
sumber
5
+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
2

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.

Charlie
sumber
0

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

crayola
sumber
-1

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;μ,σ).

A multivariate normal distribution has mean vector of length n>1 and a covariance matrix of size n×n. For two univariate random variables x,y they can be jointly distributed according to a multivariate normal distribution p(x,y)=N([x y];[μx μy],Σxy).

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;μx,σx)N(y;μy,σy).

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

JStrahl
sumber
1
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 .
Dilip Sarwate