# The Cumulative Distribution Function

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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 it cumulative distribution function.

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