# Statistics/Distributions/Bernoulli

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### Bernoulli Distribution: The coin toss

[edit | edit source]Parameters | |
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Support | |

PMF | |

CDF | |

Mean | |

Median | |

Mode | |

Variance | |

Skewness | |

Ex. kurtosis | |

Entropy | |

MGF | |

CF | |

PGF | |

Fisher information |

There is no more basic random event than the flipping of a coin. Heads or tails. It's as simple as you can get! The "Bernoulli Trial" refers to a single event which can have one of two possible outcomes with a fixed probability of each occurring. You can describe these events as "yes or no" questions. For example:

- Will the coin land
*heads*? - Will the newborn child be a girl?
- Are a random person's eyes green?
- Will a mosquito die after the area was sprayed with insecticide?
- Will a potential customer decide to buy my product?
- Will a citizen vote for a specific candidate?
- Is an employee going to vote pro-union?
- Will this person be abducted by aliens in their lifetime?

The Bernoulli Distribution has one controlling parameter: the probability of success. A "fair coin" or an experiment where success and failure are equally likely will have a probability of 0.5 (50%). Typically the variable *p* is used to represent this parameter.

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

Where the event *X=1* represents the "yes."

This distribution may seem trivial, but it is still a very important building block in probability. The Binomial distribution extends the Bernoulli distribution to encompass multiple "yes" or "no" cases with a fixed probability. Take a close look at the examples cited above. Some similar questions will be presented in the next section which might give an understanding of how these distributions are related.

#### Mean

[edit | edit source]The mean (E[X]) can be derived: