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In [[probability theory]], a ''probability density function'' (say) ''f'' is a real valued and continuous function whose value is the probability density of the variable that it is a function of. Since it is a density, the actual probability P that the variable will be in the interval [a,b] is

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The '''probability density function''' of a continuous [[random variable]] is the function that provides the likelihood that the variable will have a value in a given interval when the function is integrated over that same interval.  Stated another way, a '''probability density function''' ''f'' is a non-negative valued real function whose value is the probability density of the variable that it is a function of. Since it is a density, the actual probability P that the variable will be in the interval [a,b] is

:<math>P(a \leq x \leq b) = \int_a^b f(x) \, dx </math>

In order to qualify as a ''probability density function'', such a function must satisfy the following two criteria:

This density function is intended to express mathematically the total apportionment of the values of the variable it represents over the variables entire [[domain]]. It is necessary to express probability this way for a variable or parameter which may take on a continuum of values so that the total probability covering the entire domain of support may converge to a finite value.  The counterpart for a discretely distributed variable is the [[probability mass function]].

(1) <math> f(x) \geq 0 </math>  <math> \forall x </math> inside the domain of support.

In order to qualify, such a function must meet the following criteria:

(2) <math> \int_{-\infty}^\infty \,f(x)\,dx = 1. </math>  i.e., finitely convergent (to unity by convention).

(1) <math> \int_{-\infty}^\infty \,f(x)\,dx = 1. </math>  i.e., finitely convergent (to unity by convention).

From the first condition above, it necessarily follows that:

(2) <math> \int_{-\infty}^a \,f(x)\,dx \leq \int_{-\infty}^b \,f(x)\,dx </math>  for a<b,  i.e., is non-decreasing

<math> \int_{-\infty}^a \,f(x)\,dx \leq \int_{-\infty}^b \,f(x)\,dx </math>  for a<b,  i.e., is non-decreasing

Such a function leads to the definition of an associated [[cumulative distribution function]].

Such a function leads to the definition of an associated [[cumulative distribution function]].

If the [[domain]] of the variable is [[finite]], then the limits on the above integrals would be replaced by those bounds, and a third requirement would be:

If the [[domain]] of the variable is [[finite]], then the infinite limits on the above integrals would be replaced by those bounds, and a fourth requirement would be:

(3) <math> f(x) = 0 </math>  <math> \forall x </math>  outside the finite domain of support.

(4) <math> f(x) = 0 </math>  <math> \forall x </math>  outside the finite domain of support.

[[Category:mathematics]]

[[Category:Probability and Statistics]]