The probability density function (PDF) and the cumulative distribution function (CDF) are two related but different ways of representing the probability distribution of a random variable.
The PDF is a function that gives the probability of a random variable taking on a specific value. It is often denoted by f(x). The CDF is a function that gives the probability of a random variable being less than or equal to a specific value. It is often denoted by F(x).
The PDF is defined as the derivative of the CDF. This means that the CDF can be obtained by integrating the PDF.
The PDF is used to calculate the probability of a random variable taking on a specific value. For example, if the PDF of a random variable is f(x), then the probability of the random variable being equal to 5 is given by:
P(X = 5) = f(5)
The CDF is used to calculate the probability of a random variable being less than or equal to a specific value. For example, if the CDF of a random variable is F(x), then the probability of the random variable being less than or equal to 5 is given by:
P(X < 5) = F(5)
The PDF and the CDF are both important tools for understanding the probability distribution of a random variable. The PDF is used to calculate the probability of a specific value, while the CDF is used to calculate the probability of a range of values.
Here is a table summarizing the key differences between the PDF and the CDF: