# Posterior distribution question Pixie's question from Yahoo Answers

#### CaptainBlack

##### Well-known member

Suppose that Yi is the result of a Bernoulli trial, with probability alpha of success (Yi=1).

If we assign a Unif(0,1) prior distribution to alpha, find the posterior distribution of alpha after the observations:
(a) 1
(b) 0, 1, 1, 0, 0

Thank you so much if you can help!

#### CaptainBlack

##### Well-known member

Suppose that Yi is the result of a Bernoulli trial, with probability alpha of success (Yi=1).

If we assign a Unif(0,1) prior distribution to alpha, find the posterior distribution of alpha after the observations:
(a) 1
(b) 0, 1, 1, 0, 0

Thank you so much if you can help!
From Bayes' theorem we have:

$P(\alpha|data) = \frac{P(data|\alpha)P(\alpha)}{P(data)}$

For case (a) we have one success in one trial with prob of success $$\alpha$$ so:

$$P(data|\alpha)=\alpha$$

$$P(\alpha)=1$$ (since the prior for alpha is $$U(0,1)$$ its density is $$1$$ for $$\alpha$$ in $$[0,1]$$ and zero elswhere)

$$\displaystyle P(data) = \int_{\alpha=0}^1 P(data|\alpha)P(\alpha)\; d \alpha =\int_{\alpha=0}^1 \alpha \; d \alpha =\Bigl[ \alpha^2/2 \Bigr]_0^1=\frac{1}{2}$$

and so:
$P(\alpha|data)=2 \alpha$

For case (b) we have 2 successes in 5 trials so:

$$\displaystyle P(data|\alpha)= b(2;5,\alpha)= \frac{5!}{2!\;3! }\alpha^2 (1-\alpha)^3$$

CB