Probability density function (pdf)

If a random variable X is continuous, i.e. it may take any value within a defined range (or sometimes ranges), the probability of X having any precise value within that range is vanishingly small because a total probability of 1 must be distributed between an infinite number of values. In other words, there is no probability mass associated with any specific allowable value of X. Instead, we define a probability density function f(x) as:

i.e. f(x) is the rate of change (the gradient) of the cumulative distribution function. Since F(x) is always non-decreasing, f(x) is always non-negative.

For a continuous distribution we cannot define the probability of observing any exact value. However, we can determine the probability of lying between any two exact values (a, b):

where b > a

Example

Consider a continuous variable that is takes a Rayleigh (1) distribution. Its cumulative distribution function is given by:

and its probability density function is given by:

The probability that the variable will be between 1 and 2 is given by:

F(x) and f(x) for are plotted below: