Probability Density Function (PDF)

A probability density function (PDF) is a function that describes the probability of a random variable taking on a given value. It is a continuous function that is non-negative and integrates to 1. The PDF is often used to represent the distribution of a random variable, and it can be used to calculate the probability of the random variable taking on a given value.

The PDF is defined as follows:

$$f(x) = \frac{P(X \leq x)}{dx}$$

where $f(x)$ is the PDF, $P(X \leq x)$ is the probability that the random variable $X$ is less than or equal to $x$, and $dx$ is an infinitesimally small interval.

The PDF is a continuous function, which means that it can take on any value between 0 and 1. The PDF is also non-negative, which means that it can never be negative. Finally, the PDF integrates to 1, which means that the total probability of the random variable taking on any value is 1.

The PDF is often used to represent the distribution of a random variable. The distribution of a random variable is a description of how likely the random variable is to take on different values. The PDF can be used to calculate the probability of the random variable taking on a given value.

For example, the PDF of a normal distribution is given by the following equation:

$$f(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$

where $\mu$ is the mean of the distribution and $\sigma$ is the standard deviation.

The PDF of a normal distribution is a bell-shaped curve. The mean of the distribution is located at the center of the curve, and the standard deviation determines how wide the curve is. The probability of the random variable taking on a value within a given distance of the mean is given by the area under the curve within that distance.

The PDF is a useful tool for representing the distribution of a random variable. It can be used to calculate the probability of the random variable taking on a given value, and it can be used to compare the distributions of different random variables.