Continuous Random Variables#

Definition#

Definition 39 (Uncoutable Set)

A set \(S\) is uncountable if there is no bijection between \(S\) and the set of natural numbers \(\mathbb{N}\).

Definition 40 (Continuous Random Variables)

A continuous random variable \(X\) is a random variable1 whose cumulative distribution function (CDF) is continuous.

We can also define it via its probability density function and say that \(X\) is continuous if there is a function \(f(x)\) such that for any \(a \leq b\), we have

\[ P(a \leq X \leq b) = \int_a^b f(x) dx \]

in which case \(f(x)\) is called the probability density function (PDF) of \(X\).

We have not yet defined what the PDF and CDF are but we will do so in the next section. To be less pedantic, we can follow the definition in Definition 17 and say that \(X\) is continuous if its range is uncountably infinite. This however is not formal as stated here.

1

See Definition 14 for the definition of a random variable.