An absolutely continuous random variable is a random variable whose cumulative distribution function is a continuous function. The set of values it can take is not countable and these values, when taken one by one, have zero probability of being observed.
The adverb "absolutely" can be omitted. The term continuous random variable is a synonym of absolutely continuous random variable.
The following is a formal definition.
Definition A random variable is said to be absolutely continuous if the probability that it assumes a value in a given interval can be expressed as an integral:where the integrand function is called the probability density function of .
Note that, as a consequence of this definition, the cumulative distribution function of isBecause integrals are continuous with respect to their upper bound of integration, is continuous in , which explains why we have stated above that a continuous random variable is a random variable whose cumulative distribution function is continuous.
Let be an absolutely continuous random variable that can take any value in the interval . Let its probability density function beThen, for example, the probability that takes a value between and can be computed as follows:
Absolutely continuous random variables are discussed in more detail in the lecture entitled Random variables. You can also read a brief introduction to the probability density function, including some examples, in the glossary entry entitled Probability density function.
Next entry: Absolutely continuous random vector
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