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"Problem 3.10.7\n", | |

"\n", | |

"Poisson and binomial distributions: a student sits on a street corner for an hour and records the number of bicycles $ b $ and the number of other vehicles $v$ that go by. Two models are considered: \n", | |

"\n", | |

"* The outcomes $b$ and $v$ have independent Poisson distributions, with unknown means $\\theta_b$ and $\\theta_v$ . \n", | |

"\n", | |

"* The outcome $b$ has a binomial distribution, with unknown probability $p$ and sample size $b + v$. \n", | |

"\n", | |

"Show that the two models have the same likelihood if we define $p = \\theta_b/( \\theta_b +\\theta_v)$.\n", | |

"\n", | |

"Gelman, Andrew; Carlin, John B.; Stern, Hal S.; Dunson, David B.; Vehtari, Aki; Rubin, Donald B.. Bayesian Data Analysis, Third Edition (Chapman & Hall/CRC Texts in Statistical Science) (Page 81). CRC Press. Kindle Edition. " | |

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"This problem has no computational element, it's a fact about poisson distributions. In the first case we have\n", | |

"$$\n", | |

"P(b=b_0)=\\frac{\\theta_b^{b_0}e^{-\\theta_b)}}{b_0!}\n", | |

"$$\n", | |

"and\n", | |

"$$\n", | |

"P(v=v_0)=\\frac{\\theta_v^{v_0}e^{-\\theta_v)}}{v_0!}\n", | |

"$$\n", | |

"It's also a fact that the sum of two poisson variables with rates $\\theta_v$ and $\\theta_b$ is poisson with \n", | |

"rate $\\theta_v+\\theta_b$. \n", | |

"\n", | |

"A direct calculation gives \n", | |

"$$\n", | |

"P(b=b_0,v=v_0|b_0+v_0=N)=\\binom{N}{b_0}\\frac{\\theta_b^{b_0}\\theta_v^{N-b_0}}{(\\theta_b+\\theta_v)^{N}}\n", | |

"$$\n", | |

"which is what we're supposed to show." | |

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