Given a prior distribution $\mu$ ~ beta(a, b), what is the distribution of $\mu | X_{1}$, where $X_{1}$ is a Bernoulli random variable. What is $\mathbb{P}(X_{2} = 1 | X_{1} = 1)$?
I know from here that knowing $X_{1} = 1$, I would change the distribution to $\mu$ ~ beta(a+1, b).
Then $\mathbb{P}(X_{2} = 1) = p$, which the the parameter of the Bernoulli random variable that follows the new distribution $\mu$ ~ beta(a+1, b). I was wondering if I am right so far and how to approach further beyond just providing a beta distribution as the result here.
Any hints/thoughts are appreciated. Thanks in advance!
You have your posterior distribution given $X_1=1$ of $p \sim \text{Beta}(a+1, b)$
You then get $\mathbb P(X_2=1 \mid X_1=1) = \mathbb E[p \mid X_1=1] = \dfrac{a+1}{a+1+b}$