Consider a random vector whose entries are continuous random variables, called a continuous random vector. When taken alone, one of the entries of the random vector has a univariate probability distribution that can be described by its probability density function. This is called marginal probability density function, in order to distinguish it from the joint probability density function, which instead describes the multivariate distribution of all the entries of the random vector taken together.

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A more formal definition follows.

Definition Let be continuous random variables forming a random vector. Then, for each , the probability density function of the random variable , denoted by , is called marginal probability density function.

Recall that the probability density function is a function such that, for any interval , we havewhere is the probability that will take a value in the interval .

Instead, the joint probability density function of the vector is a function such that, for any hyper-rectanglewe havewhere is the probability that will take a value in the interval , simultaneously for all .

The marginal probability density function of is obtained from the joint probability density function as follows:In other words, the marginal probability density function of is obtained by integrating the joint probability density function with respect to all variables except .

Let be a continuous random vector having joint probability density functionThe marginal probability density function of is obtained by integrating the joint probability density function with respect to . When , thenWhen , thenTherefore, the marginal probability density function of is

Marginal probability density functions are discussed in more detail in the lecture entitled Random vectors.

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