Question

5. Polya's Urn +R • Begin with an urn containing W white balls and R red balls. Hence, N =W total balls. • Draw a random ball from the urn, check its color, and return it to the urn with another ball of the same color. • Now there are N + 1 balls in the urn. Draw 1 at random, check its color, and return it with another ball of the same color. . Now there are N + 2 balls in the urn. Draw 1 at random, check its color, and return it with another ball of the same color. ...etc. Let Rn denote the number of red balls in the urn after a total of n balls have been added. Thus, Ro=R. a) Compute E(R.:+1R = k). b) Use your answer to (a) to find an expression for E(Rm+1) in terms of E(R). c) Use your answer to (b) and induction to find E(Rn). Interpret the result. = 2m, and 6. Let 21, 22, ... be independent rolls of a standard 6-sided die, let Sn = let T = ITN=12m. a) Compute E(S.Sn = k) for m n . b) Compute E(S.Sn = k) for 1 <m<n. c) Compute E(T.TR = k) for m > n.

Answer 1

Answer: .. Then it become polls in urn Initially, N= wtR Aftea de a returning mo.of balls become N+I Then it becomes alta balls. - Ro Red balls in win After a total of n' balls in win after a total of n balls have been added and Thus, Ro=R. on ball 7 tl e Ķ L Romak Zn 6 = R, and BooB, so for every in B Rnt Bo = R + B the one adds 'c'red balls with Probability le ;P (Zn) and zero otherwise hence. El Rnt a Rnt C Zn Rome, 7 km+1) Zm K

(6) use (a) bit, fore Evern, @ R. REB p(Roth) = 2 (C) iuse [D) bit. El Rn) = 30 a R CR+B) In which is equal cors similar 9 to the probability of'R? je; E(R) - Re. As Ro=R).