# Probability Density

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## probability density

[‚präb·ə′bil·əd·ē ‚den·səd·ē]*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.

## Probability Density

The probability density of a random variable *X* is a function *p*(*x*) such that for any *a* and *b* the inequality *a* < *X* < *b* has probability equal to

For example, if *X* has a normal distribution,

If *p*(*x*) is continuous, the probability of the inequality *x* < *X* < *x* + *dx* is approximately equal to *p*(*x*) *dx* for sufficiently small *dx*. The probability density always satisfies the conditions

The probability density *p*(*x*_{1},…, *x _{s}*) for several random variables

*X*

_{1},

*X*

_{2},…,

*X*is defined in a similar manner and is called the joint probability density. Thus, for any

_{s}*a*

_{i}and

*b*

_{i}, the probability that the inequalities

*a*

_{1}<

*X*

_{1}<

*b*

_{1},…,

*a*<

_{s}*X*<

_{s}*b*are simultaneously satisfied is equal to

_{s}If the random variables *X*_{1}, *X*_{2},…, *X _{s}* have joint probability density, they will be independent if, and only if, their joint probability density is the product of the probability densities of each of them.