## Cumulative Distribution Function From a Probability Mass Function and Vice Versa

We can construct the cumulative distribution for arandom variable where a random variable may only take certain discrete values - from the probability dens function.

Suppose the probability mass function for a random variable is

0 | 1 | 2 | 3 | |

0.3 | 0.2 | 0.1 | 0.4 |

For the cumulative distribution functionm we want the values ofWe can obtain these by adding up the individual probabilities for values of up to and including We obtain

0 | 1 | 2 | 3 | |

0.3 | 0.3+0.2=0.5 | 0.3+0.2+0.1=0.6 | 0.3+0.2+0.1+0.4=1 |

Conversely, given a cumulative distribution function we can construct the probability mass function using the formula

From the cumulative distribution function above

0 | 1 | 2 | 3 | |

0.3 | 0.5-0.3=0.2 | 0.6-0.5=0.1 | 1-0.6=0.4 |

This returns the original probability mass function.