## 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 | |

0.10 | 0.05 | 0.00 | 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 | |

0.10 | 0.15 | 0.15 | 0.40 | 0.55 | 1.00 |

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 | |

0.20 | 0.25 | 0.35 | 0.40 | 0.75 | 1.00 |

is given in the table below.

0 | 1 | 2 | 3 | 4 | 5 | |

0.20 | 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 |