## Sampling Distribution for a Selection from a Discrete Distribution

In the UK coins come as 1p,2p, 5p, 10p, 20p, 50p, £1 or £2 coins. Suppose you have 5 1p coins,3 2p coins and 2 5p coins. You select three coins and are only interested in the total value of the selection. To find the sampling distribution X for the value of the selection, we must list all the possible values of the selection, along with the probability of each one.

Selection/p | Value/p X | Probability, P(X=x) |

1, 1, 1 | 3 | |

1, 1, 2 or 1, 2, 1 or 2, 1, 1 | 4 | |

1, 2, 2 or 2, 1, 2 or 2, 2, 1 | 5 | |

2, 2, 2 | 6 | |

1, 1, 5 or 1, 5, 1 or 5, 1, 1 | 7 | |

1, 2, 5 or 1, 5, 2 or 2, 1, 5 or 2, 5, 1 or 5, 1, 2or 5, 2, 1 | 8 | |

2, 2, 5 or 2, 5, 2 or 5, 2, 2 | 9 | |

1, 5, 5 or 5, 1, 5 or 5, 5, 1 | 11 | |

2, 5, 5 or 5, 2, 5 or 5, 5, 2 | 12 |

We can find the expected value of the selection using the formula

and the variance is

The variance is then