This exercise investigates properties of the Beta distribution defined in Equation (20.6).

a. Dy integrating over the range [0, 1], show that the normalization constant for the distribution beta [a, b] is given by α = Г (x + 1) = Г (a + b)/ Г (a) Г (b) where Г (x) is the Gamma function, defined by Г (x + 1) = x ∙ Г (x) and Г (1) = 1. (For integer x, Г (x + 1) = x!)

b. Show that the mean is a/ (a + b).

c. Find the mode(s) the most likely value(s) of θ).

d. Describe the distribution beta [Є, Є] for very small Є. What happens as such a distribution is updated?