1. ## beta distribution

Let P be the prob. that an MP3 player produced in a certain factory is defective, with p assumed a priori to have the uniform distribution on 0 to 1. in a sample of one hundred MP3 players, 1 is found to be defective. based on this experience, determine the posterior expected value of P.

the solution says the posterior prob. distribution of p is beta with parameters 1+1=2 and 99+1=100. i don't understand why this is beta distribution?thx.

2. From Bayesian statistics, the posterior distribution of parameter p of a binomial distribution has a beta distribution if the prior distribution of p is uniform. The parameters $\alpha-1$ is given by the number of events that happen with probability p while $\beta-1$ is given by the number of events that happen with probability 1-p.

For this question, let H denote the event that in a sample of one hundred MP3 players, 1 is found to be defective. Then,
$Pr(H|p)=\frac{100!}{99!1!}p(1-p)^{99}$
Using Bayes' theorem and total probability, we have
$f(p|H) = \frac{Pr(H|p)f(p)}{\int_0^1 Pr(H|p)f(p)dp}=\frac{\frac{100!}{99!1!}p(1-p)^{99}}{\int_0^1 \frac{100!}{99!1!}p(1-p)^{99}dp} = \frac{101!}{99!1!}p(1-p)^{99}, 0 \leq p \leq 1$
which is a beta distribution with parameters 2 and 100(and 1 and 0 for the finite support).