Cumulative Distribution Function


< Probability distributions

The cumulative distribution function (CDF) of a random variable is one way to describe the distribution of random variables.

It is an extension of a cumulative frequency table, which measures discrete counts.

However, one advantage of the CDF is that it can be defined for every type of random variable, such as discrete, continuous, or mixed. More formally, the CDF is the probability that the variable takes a value less than or equal to x.

Graph of the Cumulative Distribution Function

Graph of a cumulative distribution function, which adds probabilities as it moves to the right on the number line. The probabilities will eventually add up to 1 (or 100%).

The horizontal axis on the graph domain for the probability function. The vertical axis represents a probability, so it must fall between zero and one.

Math Behind the CDF

The cumulative distribution function (also called the distribution function) gives you the cumulative (additive) probability associated with a mathematical function. The formula will differ depending on what you are trying to calculate.

For example, the CDF for a continuous random variable is the integral

cumulative distribution function for a continuous random variable

You can use the CDF to figure out probabilities above a certain value, below a certain value, or between two values.

CDF graph: By Shailaja.k – Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=16947373