What is the probability density function?

The probability density function (PDF) is a function that describes the likelihood of a continuous random variable taking on a certain value.

In probability theory, a random variable is a variable whose value is subject to variations due to chance. A continuous random variable is one that can take on any value within a certain range, as opposed to a discrete random variable which can only take on certain values. The PDF is used to describe the probability distribution of a continuous random variable.

The PDF is defined as the derivative of the cumulative distribution function (CDF). The CDF is the probability that the random variable takes on a value less than or equal to a certain value. Mathematically, this can be written as:

F(x) = P(X ≤ x)

The PDF is then defined as:

f(x) = dF(x)/dx

In other words, the PDF is the rate of change of the CDF. The PDF can be used to calculate the probability of the random variable taking on a value within a certain range. This is done by integrating the PDF over that range. Mathematically, this can be written as:

P(a ≤ X ≤ b) = ∫a^b f(x) dx

The PDF is an important concept in probability theory and is used in many applications, including statistics, finance, and engineering.