Repeat the previous exercise for the following prior, also a particular form of the beta density:

\[p(r)=\left\{\begin{array}{cl} 2 r & 0 \leq r \leq 1 \\ 0 & \text { otherwise } \end{array}\right.\]

What are the values of the prior parameters \(\alpha\) and \(\beta\) that result in \(p(r)=2 r\) ?

Data from previous exercise

For \(\alpha, \beta=1\), the beta distribution becomes uniform between 0 and 1 . In particular, if the probability of a coin landing heads is given by \(r\) and a beta prior is placed over \(r\), with parameters \(\alpha=1, \beta=1\), this prior can be written as

\[p(r)=1 \quad(0 \leq r \leq 1)\]

Using this prior, compute the posterior density for \(r\) if \(y\) heads are observed in \(N\) tosses (i.e. multiply this prior by the binomial likelihood and manipulate the result to obtain something that looks like a beta density).