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# STAT 10-17-16

### Geometrics (Test 1)

Term | Definition |
---|---|

Continuity correction factor | use P(x<(number .5)) |

Let x~b(50, .3) | np = 15, n(1-p) = 35, both are large so normal approximation should work well |

X~b, P(x < 18) | P(x <= 17.5) |

X~b, P(x > 10) | P(x >= 10.5) |

X~b, P(9 < x <= 16) | P(9.5 <= x <= 16.5) |

X~b, P(x <= 17) | P(x <= 17.5) |

X~b, P(x >= 11) | P(x >= 10.5) |

X~b, P(11 <= x < 18) | P(11.5 <= x <= 17.5) |

Geometric distribution | binomial experiment independently repeating Bernoulli trials where we don’t know how many we perform before starting |

Geometric random variable | number of trials it takes to get first success |

Possible values of geometric distribution | 1,2,3,... |

X~g(p), P(x = 1) | P |

X~g(p), P(x = 2) | (1-p)p |

X~g(p), P(x = 3) | (1-p)^2 * p |

X~g(p), P(x = k) | (1-p)^k-1 * p |

Excel geometric distribution | no specific function, but there is a generalization |

Negative binomial random variable | number of independent Bernoulli trials needed to get the rth success |

Geometric binomial | negative binomial with r = 1 |

X~g(.8), P(x <= 1) | P(x = 1) = negbinom.dist(0,1,.8,0) |

X~g(.8), P(x < 3) | P(x <= 2) = negbinom.dist(1,1,.8,1) |

X~g(.8), P(x > 3) | 1 - P(x <= 3) = negbinom.dist(2,1,.8,1) |

X~g(.8), P(4 <= x <= 6) | P(x <= 6) - P(x <= 3) = negbinom.dist(5,1,.8,1) - negbinom.dist(2,1,.8,1) |

X~g(.6), P(x >= 2) | 1 - P(x < 2) = 1 - P(x <= 1) = 1 - .6 = .4 |

X~g(.6), P(x < 5) | P(x <= 4) = negbinom.dist(3,1,.6,1) |

X~g(.6), P(x <= 4) | negbinom.dist(3,1,.6,1) |

X~g(.6), P(3 <= x < 9) | P(x <= 8) - P(x <= 2) = negbinom.dist(7,1,.6,1) - negbinom.dist(1,1,.6,1) |

Mean: If x~g(p), then μ = | 1 / p |

Variance: If x~g(p), then σ = | (1-p)/p^2 |

Created by:
Spencer Gowey