What Is a Probability Density Function?

Probability density function (PDF) is a statistical expression that defines a probability distribution for a continuous random variable as opposed to a discrete random variable. When the PDF is graphically portrayed, the area under the curve will indicate the interval in which the variable will fall. The total area in this interval of the graph equals the probability of a continuous random variable occurring.

Understanding Probability Density Functions (PDF)

Probability density function (PDF) is not to be confused with probability mass function (PMF), which is applicable for discrete random variables. The easiest way to think about the difference between discrete vs. continuous variables is that discrete ones can be counted (i.e., the number is finite and typically takes the form of an integer) and continuous ones cannot (i.e., infinite values are possible).

Example of PDF vs. PMF

To illustrate, the probability that today's temperature will be 80 degrees—and 80 degrees exactly—is measured by PMF; the probability that the temperature will be between 80 and 85 degrees is measured by PDF. PDF calculates the probability of a range of outcomes. In the latter case, the temperature could be 80.01, 80.001, 80.0001 degrees and so on (and never reach exactly 80), or 84.99, 84.999, 84.9999, etc. (and never hit exactly 85). On a PDF graph, the probability of a single outcome is always zero. This is because a single outcome is represented by a line, which has no area under a curve. However, for continuous random variables such as the temperature range in the foregoing example, PDF can be used to calculate the probability that today's temperature will fall between 80 and 85 degrees.

PDF most commonly follows a normal distribution (Gaussian). "Black swan" or extreme outlier events are possible, but an investor who would like to estimate the probability that the S&P 500 Index will rise between 8% to 10% in 12 months, for example, can use PDF. An economist who is interested in the probability that gross domestic product (GDP) growth will land between 1% and 1.5% this quarter would also find PDF helpful.