## 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 for where may be finite or and may be finite or is given by then the cumulative distribution function, cdf, such that is given by if is continuous if X is discrete.

Example: The continuous quantity is uniformly distributed over the interval The probability distribution is The cumulative distribution function is Example: The probability distribution of a random variable is 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 the row 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 each from the previous one to give in the case of a discrete distribution.

Example: If then 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