What is probability distribution function?

A probability distribution function (PDF) is a function that describes the likelihood of a 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. The PDF of a random variable X is denoted by f(x) and is defined as the derivative of the cumulative distribution function (CDF) of X. That is, f(x) = d/dx F(x), where F(x) is the CDF of X.

The PDF of a continuous random variable is a non-negative function that integrates to 1 over the entire range of the variable. That is, ∫f(x)dx = 1. The area under the PDF curve between two values a and b represents the probability that X takes on a value between a and b. That is, P(a ≤ X ≤ b) = ∫a^bf(x)dx.

The PDF of a discrete random variable is a function that assigns probabilities to each possible value of the variable. That is, f(x) = P(X = x) for all possible values of x. The sum of the probabilities of all possible values of X is 1. That is, ∑f(x) = 1.

The PDF is a fundamental concept in probability theory and is used to model a wide range of phenomena, from the distribution of heights in a population to the distribution of stock prices in financial markets. Understanding the PDF is essential for many areas of mathematics and statistics, including hypothesis testing, estimation, and regression analysis.