Cantor distribution
= Cantor distribution
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{wiki}
The <Cantor distribution> is the <law of a random variable>
$$
X=\sum_{n=1}^{\infty}\frac{2B_n}{3^n},
$$
where the $B_n$ are <independent> <Bernoulli random variables> taking values zero and one with equal probability. Its values lie in the <Cantor set>, and its <cumulative distribution function> is the <Cantor function>. Every individual value has probability zero, but the <Cantor set> has probability one despite having zero <Lebesgue measure>. A <probability density function> on that set would integrate to zero, so it cannot describe this law. The example motivates the distinction between a <continuous probability distribution> and an <absolutely continuous probability distribution>.