Suppose that $X_{1}, \ldots, X_{n}$ are i.i.d. coin tosses with probability $\theta$ of obtaining a head.

(a) Compute the posterior distribution of $\theta$ given the observations $X_{1}, \ldots, X_{n}$ in the case of a uniform prior on $[0,1]$.

(b) Give the definition of the quadratic error loss function.

(c) Determine the value $\widehat{\theta}$of $\theta$ which minimizes the quadratic error loss function. Justify your answer. Calculate $\mathbb{E}[\hat{\theta}]$.

[You may use that the $\beta(a, b), a, b>0$, distribution has density function on $[0,1]$ given by

$$c_{a, b} x^{a-1}(1-x)^{b-1}$$

where $c_{a, b}$ is a normalizing constant. You may also use without proof that the mean of a $\beta(a, b)$ random variable is $a /(a+b) .]$