Probability density function

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Two Probability Density Functions

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

P(a \leq x \leq b) = \int_a^b f(x) \, dx

This density function is intended to express mathematically the total apportionment of the values of the variable it represents over the variables entire domain. This "apportionment" can signify different things in various contexts, such as relative proportion of observations, or information regarding the residual uncertainty of its true value.

It is necessary to express probability as a density function 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.

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

(1)  f(x) \geq 0  \forall x inside the domain of support.

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

From the first condition above, it necessarily follows that:

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

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

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:

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

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