If we observe X = 1, what is the posterior distribution of the parameter theta?
The posterior probability density is given by Bayes' rule:
p(θ | X) = p(θ) * P(X | θ) / P(X) = 1 * θ / 0.5 = 2θ
Note: The fact that P(X) = 0.5 could have been calculated using the method in part (b), or more simply by considerations of symmetry: since θ is uniformly distributed, X is equally likely to be 0 or 1.