Distribution Functions

Cumulative Distribution Function

Definition (Cumulative distribution function)

The distribution is often given in terms of the distribution function, or cumulative distribution function (CDF), which is the function , defined by

The CDF effectively contains all the information about the random variable.

Theorem

Let have CDF and let have CDF . If for all , then for all .

Note that satisfies the following:

• is a non-decreasing function
• is right continuous, i.e, for every , F(x)

Probability Mass Function

Definition (Probability mass function)

is discrete if it takes countably many values. The probability function or probability mass function (PMF) of a discrete random variable is the function .

The PMF must satisfy that for all , and that . The CDF of is related to by

Probability Density Function

Definition (Probability density function)

A random variable is continuous if there exists a function such that for all , , and for every ,

The function is called the probability density function (PDF). If the density exists, then we have that

and at all points at which is differentiable.

A density satisfies

Note that if is continuous, then for every . It is not valid to think of as ; only for discrete random variables. We get probabilities from PDF by integrating. A PDF can be bigger than 1, unlike the mass function.

Lemma

Let be the CDF for a random variable . Then:

Quantile Function

Definition (Quantile function)

Let be a random variable with CDF . The inverse CDF or quantile function is defined by

for . If is strictly increasing and continuous, then is the unique real number such that .

Sources

• Wasserman, L. (2010). All of Statistics: A concise Course in Statistical Inference. Chapter 2.