$ \theta $ is uniform [0,1]
supp. $ \theta $ = 1/3
MSE = E[ ($ \theta $ - 1/3)^2 ] = E[ $ \theta $^2 - 2*(1/3)*$ \theta $ + (1/3)^2 ] = 1/3 - 2(1/3)(1/2) + (1/3)^2 = (1/3)^2 = 1/9
supp. instead $ \theta $ = 1/2
MSE = E[ ($ \theta $ - 1/2)^2 ] = E[ $ \theta $^2 - 2*(1/3)*$ \theta $ + (1/4) ] = 1/3 - 1/2 + 1/4 = 1/12
