probability density; probability density function

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For a continuous random variable X the probability density function f is such that

for all x₁<x₂. Because, for a continuous random variable, P(X=x_j)=0 for any value x_j, either or both of the '<' signs in the left-hand side can be replaced with '≤'. If the interval of possible values for X is (a, b), then


It is often convenient to regard the function f as being defined for all real values of x. This can be achieved by taking f(x)=0 for x outside the interval (a, b), so that


This property, and the property that f(x)≥0 for all x, are essential properties of a probability density function. It should be noted that, although f(x) is related to probability, f(x) can exceed 1.

The probability density function is often referred to simply as the probability density. It is related to the cumulative distribution function F by


The probability density function is not necessarily continuous. At a point of discontinuity the value assigned to f(x) is immaterial.

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