Cumulative Distribution Function

probability-theory

Definition

Cumulative Distribution Function

The cumulative distribution function of a random variable (continuous or discrete) is defined as:

Properties:

• is bounded, i.e.:
• is monotonically increasing, i.e.:
• is right-continuous:
• Left/right asymptotes are:

Discrete vs Continuous

Discrete

The cumulative distribution function is a step function for discrete random variable.

Let be a discrete random variable taking values with pmf and cumulative distribution function , then:

From that, we get:

Continuous

Let be a continuous random variable with density and cumulative distribution function , then:

From that, we get:

and the density is the cumulative distribution function’s derivative:

Inverse

One often needs the inverse function of a cdf . Per definition, is monotonically increasing, however, there’s no guarantee that is injective.

Therefore, we need the generalised inverse of .