Question

Exercise 4.4. Liz is standing on the real number line at position 0. She rolls a die repeatedly. If the roll is 1 or 2, she takes one step to the right (in the positive direction). If the roll is 3, 4, 5 or 6, she takes two steps to the right. Let Xn be Liz's position after n flips of the coin. Estimate the probability that X90 is at least 160 Section 4.2 Exercise 4.5. Consider the setup of Exercise 4.4. Find the limits below and explain your answer. 1.6n) (a) Find lim P(X n o0 (b) Find lim P(X 1.7n) > n-o0
DO 4.5 correctly, DO NOT copy answer from other ones, the answer to 4.5 is provided as part a) 1 and b)0

Answer 1

Answers Given that 2:=f' with probability &=5 { with probability 4-2 Elzi) = 1 * ()+2*() 21.67 ELE=)=15*15) +25 +13) § Vous (21) = €128)- € (21) = 3-11.6753 23- 2.7889 0.210 Xn = F zi, E(Xn) = 1.67n, Vasl xn)= 0.211in .

Xn -1.67n -> N(0,1), (0.24 in) as ndo (apply central limits theorem). (a) P(xn >1.6n). I Xo - 1.67n lign-1.ban Pen >1.6n) = P(aquin > aliin! :1-(-0.03316) 2010.3316) 20:6299 SO PCxo > l6n) - 0:6299 b) P(xn> 167n) .pl xn-1.671. lian-1.63 0.211n 0.2iin -|- 0 (0.1421) 20:4435 P(xn > 1.70) -0.4435.