Probability density function (pdf) | Vose Software

Probability density function (pdf)

See also: The basics of probability theory introduction, Probability mass function (pmf), Cumulative distribution function (cdf)

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


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:



Read on: Cumulative distribution function (cdf)




Monte Carlo simulation in Excel. Learn more

Spreadsheet risk analysis modeling


Adding risk and uncertainty to your project schedule. Learn more

Project risk analysis


Enterprise Risk Management software (ERM)

Learn more about our enterprise risk analysis management software tool, Pelican

Enterprise risk management software introduction