## The Cumulative Distribution Function

A probability distribution is usually defined in terms of it's probability distribution function (if continuous), the probability that it takes a value in a certain range, or probability mass function (if discrete), the probability that it takes a certain value. Sometimes it is more convenient to define it in terms of a cumulative distribution function.

If the probability density function forwheremay be finite orandmay be finite oris given bythen the cumulative distribution function, cdf, such thatis given by

ifis continuous

if X is discrete.

Example: The continuous quantityis uniformly distributed over the intervalThe probability distribution isThe cumulative distribution function is

Example: The probability distribution of a random variableis given in the following table. Construct the cumulative distribution function.

 0 1 2 3 4 5 0.1 0.05 0 0.25 0.15 0.45

To find the cumulative distribution function, add up the entries in therow as you go along, to give

 0 1 2 3 4 5 0.1 0.15 0.15 0.4 0.55 1

Conversely given a cumulative distribution function we can find the probability distribution function by differentiation, or by subtraction eachfrom the previous one to givein the case of a discrete distribution.

Example: Ifthen

Example:is given in the following table.

 0 1 2 3 4 5 0.2 0.25 0.35 0.4 0.75 1

is given in the table below.

 0 1 2 3 4 5 0.2 0.25-0.20=0.05 0.35-0.25=0.10 0.40-0.35=0.05 0.75-0.40=0.35 1.00-0.75=0.25