## The Cumulative Distribution Function

A probability distribution is usually defined in terms of it'sprobability distribution function (if continuous), the probabilitythat it takes a value in a certain range, or probability massfunction (if discrete), the probability that it takes a certain valueSometimes it is more convenient to define it in terms of itcumulative distribution function.

If the probability density function forwheremaybe finite orandmaybe finite orisgiven bythenthe cumulative distribution function, cdf, such thatis given by

ifiscontinuous

ifX is discrete.

Example: The continuous quantityisuniformly distributed over the intervalTheprobability distribution isThecumulative distribution function is

Example: The probability distribution of a random variableisgiven in the following table. Construct the cumulative distributionfunction.

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 entriesin therowas 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 cumulativedistribution function we can find the probability distributionfunction by differentiation, or by subtraction eachfromthe previous one to giveinthe case of a discrete distribution.

Example: Ifthen

Example:isgiven in the following table.

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

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

isgiven 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 |