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随机偏置硬币

Coin-making machine

专题
Probability / 概率
难度
L4

题目详情

概率题：随机偏置硬币：观测后 PP 的后验分布。

英文原题

A coin- making machine produces pennies. Each penny is manufactured to have a probability PP of turning up heads. However, the machine draws PP randomly from the uniform distribution on [0,1][0,1] so PP can differ for each coin produced. A coin pops out of the machine. You flip it once, and it comes up heads. Given this information, what is the (conditional) distribution function FP∣H(p)F_{P|H}(p) for the probability of a head for that coin (where " HH " denotes conditioning on the head)?

What is the (conditional) distribution function for the probability of a head if you flip the coin 1,000 times and get 750 heads?

解析

先验 P∼Unif(0,1)P\sim\mathrm{Unif}(0,1) 等价于 P∼Beta(1,1)P\sim\mathrm{Beta}(1,1)。

  • 只抛 1 次且结果为正面（H）：后验
P∣H∼Beta(2,1),P\mid H\sim\mathrm{Beta}(2,1),

因此后验分布函数为

FP∣H(p)=p2, 0≤p≤1.\boxed{F_{P\mid H}(p)=p^2,\ 0\le p\le 1}.
  • 抛 1000 次得到 750 个正面：后验
P∣data∼Beta(751,251).\boxed{P\mid \text{data}\sim\mathrm{Beta}(751,251)}.

其分布函数为不完全 Beta 函数的正则化形式：

FP∣data(p)=Ip(751,251).F_{P\mid \text{data}}(p)=I_p(751,251).

英文解析

P∣H∼Beta(2,1),P\mid H\sim\mathrm{Beta}(2,1), FP∣H(p)=p2, 0≤p≤1.\boxed{F_{P\mid H}(p)=p^2,\ 0\le p\le 1}. P∣data∼Beta(751,251).\boxed{P\mid \text{data}\sim\mathrm{Beta}(751,251)}. FP∣data(p)=Ip(751,251).F_{P\mid \text{data}}(p)=I_p(751,251).