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(i) Let RW, = {(so.S1.. sn) : So = 0, Isi+1 – Si| = 1,i = 0, 1,...,n– 1} be the set of all possible paths of a simple random walk of length n. Show that #RW, = 2", or the number of paths in RW, is 2". (ii) Let Dn = {(So, S1,.., Sn) € RW, : Sn = 0}. Calculate the number of paths in D,?

(i) Let RW, = {(so.S1.. sn) : So = 0, Isi+1 – Si| = 1,i = 0, 1,...,n– 1} be the set of all possible paths of a simple random walk of length n. Show that #RW, = 2", or the number of paths in RW, is 2". (ii) Let Dn = {(So, S1,.., Sn) € RW, : Sn = 0}. Calculate the number of paths in D,?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Execise. (i) Let RW, = {(so, $1,..., Sn) : so = 0, |Si+1 – s;l = 1,i = 0, 1,...,n – 1} %3D be the set of all possible paths of a simple random walk of length n. Show that #RW, = 2", or the number of paths in RW, is 2". (ii) Let D, = {($o, S1,..., Sn) € RW, : Sn = 0}. %3D Calculate the number of paths in Dn?
Transcribed Image Text:Execise. (i) Let RW, = {(so, $1,..., Sn) : so = 0, |Si+1 – s;l = 1,i = 0, 1,...,n – 1} %3D be the set of all possible paths of a simple random walk of length n. Show that #RW, = 2", or the number of paths in RW, is 2". (ii) Let D, = {($o, S1,..., Sn) € RW, : Sn = 0}. %3D Calculate the number of paths in Dn?
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