I am trying to solve this problem: A new drug has p chance of working, where p is a random variable with density $f_p(a)\sim Beta(2,3)$. They test the drug on five animals, three of whom show that the drug works. What is the posterior distribution on $p$ given this information?
I found the beta density as: $2p(1−p)2⋅\mathbb(p \in [0,1]$ But I am not sure where to begin?
Can I just add $(2,3)+(3,2)$ to get $Beta(5,5)$?
The prior is
$$\pi(\theta)\propto \theta(1-\theta)^2$$
The likelihood is
$$p(\mathbf{x}|\theta)\propto \theta^3(1-\theta)^2$$
Multiplying you get
$$\pi(\theta|\mathbf{x})\propto \theta^4(1-\theta)^4$$
Thus
$$\pi(\theta|\mathbf{x})\sim Beta(5;5)$$