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.

The following is a formal definition.

Definition The probability density function of a continuous random variable is a function such thatfor any interval .

The set of values for which is called the support of .

Suppose a random variable has probability density function

To compute the probability that takes a value in the interval ,you need to integrate the probability density function over that interval:

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.

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

Previous entry: Prior probability

Next entry: Probability mass function

The book

Most learning materials found on this website are now available in a traditional textbook format.

Featured pages

- Permutations
- Binomial distribution
- Maximum likelihood
- Poisson distribution
- Moment generating function
- Bernoulli distribution

Explore

Main sections

- Mathematical tools
- Fundamentals of probability
- Probability distributions
- Asymptotic theory
- Fundamentals of statistics
- Glossary

About

Glossary entries

Share