Consider the Polya urn model. (a) Show that the density of is indeed the Beta-density
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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 Wn be the proportion of red balls in the urn after the nth drawing. Compare the limiting distributions of Wn and X̅n.
(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.
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