Let X₁, X₂,... be independent random variables, each taking a value of either 1 (with probability p) or −1 (with probability q = 1 − p). Assume So= = a for some a EZ and let Sn = a + X₁ for n ≥ 1. i=1 Sn is known as the simple random walk. Show that: (i) Sn is spatially homogenous: n P(Sn = b|So = a) = P(Sn = b +c|So = a + c). (ii) Sn is temporally homogenous: P(Sn = b|So = a) = P(Sm+n = b[Sm = a). (iii) Sn has the Markov property: P(Sm+n=b|So = a0, S₁ = a1,..., Sm = Note: |ai — ai_1] = 1 for 1 ≤ i < m am) = P(Sm+n = b|Sm = am).
Let X₁, X₂,... be independent random variables, each taking a value of either 1 (with probability p) or −1 (with probability q = 1 − p). Assume So= = a for some a EZ and let Sn = a + X₁ for n ≥ 1. i=1 Sn is known as the simple random walk. Show that: (i) Sn is spatially homogenous: n P(Sn = b|So = a) = P(Sn = b +c|So = a + c). (ii) Sn is temporally homogenous: P(Sn = b|So = a) = P(Sm+n = b[Sm = a). (iii) Sn has the Markov property: P(Sm+n=b|So = a0, S₁ = a1,..., Sm = Note: |ai — ai_1] = 1 for 1 ≤ i < m am) = P(Sm+n = b|Sm = am).
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Transcribed Image Text:Let X₁, X₂,... be independent random variables, each taking a value of either 1 (with
probability p) or -1 (with probability q = 1 − p). Assume So = a for some a € Z and
let
Sn is known as the simple random walk. Show that:
(i) Sn is spatially homogenous:
(ii) Sn is temporally homogenous:
Sn = a + X₁ for n ≥ 1.
Note: |ai-ai-1|
(iii) Sn has the Markov property:
=
n
P(Sn = b|So = a) = P(Sn = b + c|So = a + c).
i=1
P(Sn = b|So = a) = P(Sm+n = b|Sm = a).
P(Sm+nb|So ao, S₁ = a₁,..., Sm=am) = P(Sm+n=b|Sm= am).
=
1 for 1 < i <m.
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