Question

Suppose a particle moves along the x-axis beginning at 0. It moves one integer step to the left or right with equal probability.

a)What is the pdf of its position after three steps?

b)What is the pdf of its position after three steps if the x-axis beginning at 1?

c)What is the pdf of its position after two steps?

d)What is the pdf of its position after two steps if the x-axis beginning at 1?

Answer 1

a) We are given here for each step, there is a 0.5 probability that it moves right and 0.5 probability that it moves left.

After 3 steps the following possibilities are there:

• RRR, +3, Prob = 0.5³ = 0.125
• RRL, RLR, LRR, +1, Prob = 3*0.5³ = 0.375
• LLR, LRL, RLL, -1, Prob = 3*0.5³ = 0.375
• LLL, -3, Prob = 0.5³ = 0.125

The PDF of its position after 3 steps is this given as:

P(X = -3) = 0.125,
P(X = -1) = 0.375,
P(X = 1) = 0.375,
P(X = 3) = 0.125

b) Given that the beginning spot was 1, the PDF after 3 steps here would be given as:

P(X = -2) = 0.125,
P(X = 0) = 0.375,
P(X = 2) = 0.375,
P(X = 4) = 0.125

c) After 2 steps, the possibilities are obtained as:

• RR, +2, Prob = 0.25
• LR and RL, 0, Prob = 2*0.25 = 0.5
• LL, -2, Prob = 0.25

Therefore PDF here is given as:

P(X = -2) = 0.25
P(X = 0) = 0.5
P(X = 2) = 0.25

d) Here, again as we have to add 1, we get the PDF here as:

P(X = -1) = 0.25
P(X = 1) = 0.5
P(X = 3) = 0.25