Random Walk Models

Question 1

If P(Zn = 1) = p, P(Zn = -1) = q, Xn = x0 + sum(k=1 to n) Zk, what is E(Zn) and proof?

Answer 1

E(Zn) = p(1) + q(-1) = p - q

Question 2

If P(Zn = 1) = p, P(Zn = -1) = q, Xn = x0 + sum(k=1 to n) Zk, what is Var(Zn) and proof?

Answer 2

Var(Zn) = 4pq

Question 3

If P(Zn = 1) = p, P(Zn = -1) = q, Xn = x0 + sum(k=1 to n) Zk, what is E(Xn) and proof?

Answer 3

E(Xn) = x0 + n(p-q)

Question 4

If P(Zn = 1) = p, P(Zn = -1) = q, Xn = x0 + sum(k=1 to n) Zk, what is Var(Xn) and proof?

Answer 4

Var(Xn) = 4pqn

Question 5

If P(Zn = 1) = p, P(Zn = -1) = q, Xn = x0 + sum(k=1 to n) Zk, normalise

Answer 5

[1 / sqrt(4npq)] * (Xn - E(Xn))

Question 6

If p≠q, what is the probability of absorption (ruin)?

Answer 6

fk = [ (p/q)^(a-k) - 1 ] / [ (p/q)^a - 1 ]

Question 7

If p=q, what is the probability of absorption (ruin)?

Answer 7

fk = 1 - k/a

Question 8

If p≠q, what is the probability of absorption (win)?

Answer 8

wk = [ 1 - (q/p)^k ] / [ 1 - (q/p)^a ]

Question 9

If p=q, what is the probability of absorption (win)?

Answer 9

wk = k/a

Question 10

What is the expected duration of a walk if p≠q?

Answer 10

dk = k/(q-p) - a/(q-p) * [ (1 - (q/p)^k) / (1 - (q/p)^a) ]

Question 11

What is the expected duration of a walk if p=q?

Answer 11

dk = k(a-k)

Question 12

What does the probability of absorption (win) become if there is only one barrier?

Answer 12

if q<p></p>

Question 13

What is the expected duration of a walk become if there is only one barrier?

Answer 13

if q≤p, dk = infinity

if q>p, dk = k / (q-p)