Probability density function

A probability density function (PDF) is a mathematical tool used to describe continuous random variables. In probability theory, a continuous variable can take on an infinite set of possible values. Because there are infinite values, the absolute likelihood of a variable equaling one exact, specific point is zero. Instead, the PDF provides a way to understand relative likelihood. It tells us how much more likely a variable is to fall near one value compared to another.

The mechanism of a PDF relies on the concept of density per unit. You can think of the value of the PDF at a specific point as the probability per unit length. To find the actual probability of an event, you must look at a range of values rather than a single point. This is done by calculating the integral of the function over that specific range. The resulting probability is represented by the non-negative area under the curve between the lowest and highest values of the range.

There are distinct types of functions used depending on the nature of the data. A probability mass function (PMF) is used for discrete random variables. These are variables that take values on a countable set, like counting individual objects. In contrast, the PDF is used for continuous random variables, such as time or height. While some sources use the term "probability distribution function" to refer to a PDF, this can be confusing. In many contexts, that term might instead refer to a cumulative distribution function (CDF) or a PMF.

Historically and mathematically, the relationship between these functions is very precise. If a distribution has a density, the probability of every one-point set is zero. This is because the probability is tied to the area under the curve, and a single point has no width. A distribution has a density function if its cumulative distribution function is absolutely continuous. In these cases, the PDF is the derivative of the cumulative distribution function. This mathematical connection allows scientists to move between total accumulated probability and local density.

Specific numbers and properties define how these functions behave in the real world. The PDF must be non-negative everywhere, meaning the curve never drops below the horizontal axis. The total area under the entire curve must always equal exactly one. This represents a 100% probability that the variable will fall within the set of all possible values. Interestingly, the value of the PDF itself can be greater than one. For example, a continuous uniform distribution can have a density value higher than one if the interval is very small.

Consider the example of bacteria life spans to deepen your understanding. If a species of bacteria typically lives between 20 and 30 hours, the probability of one dying at exactly 5 hours is zero. However, the probability of it dying between 5 and 5.01 hours might be 0.02, or 2%. If you shorten that window to 5 to 5.001 hours, the probability drops to 0.002. The ratio of the probability to the duration of the interval remains constant. In this case, the density is 2 per hour.

Probability density functions also connect to broader, more complex systems. When studying multiple variables at once, mathematicians use a joint probability density function. This describes the probability of a set of variables falling within a specific multi-dimensional domain. From a joint density, you can derive a marginal density function. This is done by integrating over the values of the other variables to focus on just one. This allows researchers to study how different parts of a complex system relate to one another or act independently.