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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

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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.


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

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

The probability density is not a probability

It is important to understand a fundamental difference between the probability density function, which characterizes the distribution of a continuous random variable, and the probability mass function, which characterizes the distribution of a discrete random variable (remember: a random variable is discrete if the number of values it can take is countable, while the number of values that a continuous random variable can take is uncountable). The probability mass function of a discrete variable Y is a function [eq7] that gives you, for any real number $y$, the probability that Y will be equal to $y$. On the contrary, if X is a continuous variable, its probability density function [eq8] evaluated at a given point x is not the probability that X will be equal to x. As a matter of fact, this probability is equal to zero for any x because[eq9]where [eq10] is any primitive (or indefinite integral) of [eq11].

If you are puzzled by the latter result, you are advised to read the lecture on zero-probability events.

Although it is not a probability, the value of the pdf at a given point x can be given a straightforward interpretation:[eq12]where $Delta x$ is a small increment.


The proof we are going to give is not rigorous. Rather, we are focusing on the intuition. For the sake of simplicity, we assume that the pdf is a continuous function. Strictly speaking, this is not necessary, although most of the pdfs that are encountered in practice are continuous (by definition, a pdf must be integrable; however, while all continuous functions are integrable, not all integrable functions are continuous). If the pdf is continuous and $Delta x$ is small, then [eq13] is well approximated by [eq14] for any $t$ belonging to the interval [eq15]. It follows that [eq16]

In the above approximate equality, we consider the probability that X will be equal to x or to a value belonging to a small interval near x. In particular, we consider the interval [eq17]. The probability is proportional to the length $Delta x$ of the small interval we are considering. The constant of proportionality [eq14] is the probability density function of X evaluated at x. Thus, the higher the pdf [eq14] is at a given point x, the higher is the probability that X will take a value near x.

Related concepts

Related concepts are those of:

More details

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

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