Consider the Polya urn model. (a) Show that the density of is indeed the Beta-density from (4.2.4). In particular, make sure that E{}

Consider the Polya urn model.

(a) Show that the density of Θ is indeed the Beta-density from (4.2.4). In particular, make sure that E{Θ} indeed equals r/b+r. (Advice: Proceeding from the formula Γ(t +1) = tΓ(t), show that the r.-h.s. of (4.2.3) equals

Using the formula

find the moments of the Beta distribution. Compare what you got.)

(b) Let W_n be the proportion of red balls in the urn after the nth drawing. Compare the limiting distributions of W_n and X̅_n.

(c) Find these distributions for the case r = b = c; for instance, when in the beginning, there are one red and one black ball, and each time one ball is added.