Consider a continuous random vector, whose entries are continuous random variables.
Each entry of the random vector has a univariate distribution described by a probability density function (pdf). This is called marginal probability density function, to distinguish it from the joint probability density function, which depicts the multivariate distribution of all the entries of the random vector.
A more formal definition follows.
Definition Let be continuous random variables forming a continuous random vector. Then, for each , the pdf of the random variable , denoted by , is called marginal probability density function.
Before explaining how to derive the marginal pdfs from the joint pdf, let us revise the basics of pdfs.
The probability density function of a variable is a function such that the probability that will take a value in the interval isfor any interval
The joint probability density function of the vector is a function such that the probability that will take a value in the interval , simultaneously for all , isfor any hyper-rectangle
The marginal probability density function of is obtained from the joint pdf as follows:
In other words, to compute the marginal pdf of , we integrate the joint pdf with respect to all the variables except .
Let be a continuous random vector having joint pdf
To derive the marginal probability density function of , we integrate the joint pdf with respect to .
When , then
When , then
Therefore, the marginal probability density function of is
On the following pages you can find other examples and detailed derivations:
the marginals of a multivariate normal density are univariate normals;
the marginals of a multivariate Student density are univariate t;
Marginal probability density functions are discussed in more detail in the lecture on Random vectors.
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Please cite as:
Taboga, Marco (2021). "Marginal probability density function", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/glossary/marginal-probability-density-function.
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