Cumulative distribution function

A cumulative distribution function $𝐹 𝑋 : ℝ \to ℝ$ (CDF) may be defined for either a Discrete random variable or a Continuous random variable such that

𝐹 𝑋 (𝑡) = ℙ (𝑋 \leq 𝑡)

This can be used to find the probability of any range

ℙ (𝑎 \leq 𝑋 \leq 𝑏) = 𝐹 𝑋 (𝑏) - 𝐹 𝑋 (𝑏)

The cumulative distribution function has the following properties

$0 \leq 𝐹 𝑋 (𝑡) \leq 1$ which follows from its definition in the context of the Probability model.
$𝐹 𝑋$ is monotone increasing, i.e. $𝑡 1 \leq 𝑡 2 ⟹ 𝐹 𝑋 (𝑡 1) \leq 𝐹 𝑋 (𝑡 2)$ , see below.
$l i m 𝑡 \to \infty 𝐹 𝑋 (𝑡) = 1$ , see below.
$l i m 𝑡 \to - \infty 𝐹 𝑋 (𝑡) = 0$ , see below.
Continuity varies between the discrete and continuous case, see below.

Random variables sharing the same CDF form equivalence classes under $\sim$ . See Variable distribution equivalence.

A useful property of the CDF is that given any variable $𝑋$ with CDF $𝐹 𝑋$

𝐹 𝑋 \circ 𝑋 \sim U (0, 1)

Discrete variable

For a Discrete random variable, the cumulative distribution function is defined using the probability mass function $𝑝 𝑋 (𝑥)$ such that

𝐹 𝑋 (𝑡) = \sum 𝑥 \leq 𝑡 𝑝 𝑋 (𝑥)

For a discrete variable $𝐹$ is right-continuous.

For a Continuous random variable, the cumulative distribution function is defined using the probability density function $𝑓 𝑋 (𝑥)$ such that

𝐹 𝑋 (𝑡) = \int 𝑡 - \infty 𝑓 𝑋 (𝑥) 𝑑 𝑥

For a continuous variable $𝐹$ is continuous.

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