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Probability density function

The distribution of a continuous random variable can be characterized through its probability density function (pdf). The probability that a continuous random variable takes a value in a given interval is equal to the integral of its probability density function over that interval, which in turn is equal to the area of the region in the xy-plane bounded by the x-axis, the pdf and the vertical lines corresponding to the boundaries of the interval.

For example, in the picture below the blue line is the pdf of a normal random variable and the area of the red region is equal to the probability that the random variable takes a value comprised between -2 and 2.

Probability density function of a normal distribution

Definition

The following is a formal definition.

Definition The probability density function of a continuous random variable X is a function [eq1] such that[eq2]for any interval [eq3].

The set of values x for which [eq4] is called the support of X.

Example

Suppose a random variable X has probability density function[eq5]

To compute the probability that X takes a value in the interval $\left[ 1,2\right] $,you need to integrate the probability density function over that interval:[eq6]

Related concepts

Related concepts are those of joint probability density function, which characterizes the distribution of a continuous random vector, marginal probability density function, which characterizes the distribution of a subset of entries of a random vector, and conditional probability density function, which is a pdf obtained by conditioning on the realization of another random variable.

More details

Probability density functions are discussed in more detail in the lecture entitled Random variables.

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Next entry: Probability mass function

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