1515 0935 POZ 60 576 UNIT 4 Random Walk Larry Gui Aug 20is 6

1.515) 0.935 POZ 60 57.6 UNIT 4 Random Walk Larry Gui Aug 20is 64 Exam Revision Questions Ouestion #1 In an unrestricted random walk, the probability that a particle moves 2 steps to the right is p, and 1 step to the left is q 1-p. Xn is the random variable representing the position of the particle to the right of the starting position after n moves. (a) List all the possible sample paths taken by the particle that leads to (i) X E 1; (ii) X 2. (b) Hence express in terms of p and q (i) P (X 1); (ii) P(X5 2)

Solution

a1.

What matters is that there are 3 steps to the right and 2 to the left. Thus,

RRRLL
RRLRL
RRLLR
RLRRL
RLRLR
RLLRR
LRRRL
LRRLR
LRLRR
LLRRR

A2.

It is impossible to get x5 = 2. There are no such possibilities.

*********************

b1.

Using binomial probability,

P(X5 = 1) = 5C3 p^3 q^2 [ANSWER]

********************

B2.

As this case is impossible,

P(X5 = 2) = 0 [ANSWER]