Let X ∼ Beta

(a, b).

(a) For a > 2 and b > 2, show that the mode of X is given by x = a−1 a+b−2 .

(b) Write an equation in x for the median of X when a = 3 and b = 2 and verify that x ≈ 0.614272.

(c) Find a and b such that X is symmetric with a standard deviation of 0.2.

(d)  If I believe that X ∼ Beta

(a,

b) with mode at 5/8 and a standard deviation of approximately 0.1, what integer values of parameters a and b could I use?