Continuous random variable

By one convention, a probability distribution is called continuous if its cumulative distribution function is continuous. That is equivalent to saying that for random variables X with the distribution in question, Pr[X = a] = 0 for all real numbers a, i.e.: the probability that X attains the value a is zero, for any number a.

While for a discrete probability distribution one could say that an event with probability zero is impossible, this can not be said in the case of a continuous random variable, because then no value would be possible.

This paradox is solved by realizing that the probability that X attains a value in an uncountable set (for example an interval) can not be found by adding the probabilities for individual values.

By another convention, the term "continuous probability distribution" is reserved for distributions that have probability density functions. These are most precisely called absolutely continuous random variables (see Radon–Nikodym theorem).

A random variable with the Cantor distribution is continuous according to the first convention, but according to the second, it is not (absolutely) continuous. Also, it is not discrete nor a weighted average of discrete and absolutely continuous random variables.

In practical applications random variables are often either discrete or absolutely continuous.

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