# Statistics/Distributions/Geometric

### Geometric Distribution

[edit | edit source]Probability mass function | |

Cumulative distribution function | |

Parameters | success probability (real) |
---|---|

Support | |

PMF | |

CDF | |

Mean | |

Median | (not unique if is an integer) |

Mode | |

Variance | |

Skewness | |

Ex. kurtosis | |

Entropy | |

MGF | , for |

CF |

There are two similar distributions with the name "Geometric Distribution".

- The probability distribution of the number
*X*of Bernoulli trials needed to get one success, supported on the set { 1, 2, 3, ...}

- The probability distribution of the number
*Y*=*X*− 1 of failures before the first success, supported on the set { 0, 1, 2, 3, ... }

These two different geometric distributions should not be confused with each other. Often, the name *shifted* geometric distribution is adopted for the former one. We will use *X* and *Y* to refer to distinguish the two.

#### Shifted

[edit | edit source]The shifted Geometric Distribution refers to the probability of the number of times needed to do something until getting a desired result. For example:

- How many times will I throw a coin until it lands on
*heads*? - How many children will I have until I get a girl?
- How many cards will I draw from a pack until I get a Joker?

Just like the Bernoulli Distribution, the Geometric distribution has one controlling parameter: The probability of success in any independent test.

If a random variable X is distributed with a Geometric Distribution with a parameter p we write its probability mass function as:

With a Geometric Distribution it is also pretty easy to calculate the probability of a "more than n times" case. The probability of failing to achieve the wanted result is .

Example: a student comes home from a party in the forest, in which interesting substances were consumed. The student is trying to find the key to his front door, out of a keychain with 10 different keys. What is the probability of the student succeeding in finding the right key in the 4th attempt?

#### Unshifted

[edit | edit source]The probability mass function is defined as:

- for

#### Mean

[edit | edit source]Let *q=1-p*

We can now interchange the derivative and the sum.

#### Variance

[edit | edit source]We derive the variance using the following formula:

We have already calculated E[*X*] above, so now we will calculate E[*X ^{2}*] and then return to this variance formula:

Let *q=1-p*

We now manipulate *x ^{2}* so that we get forms that are easy to handle by the technique used when deriving the mean.

We then return to the variance formula