Lecture01rnd (PDF)




File information


This PDF 1.5 document has been generated by TeX / MiKTeX pdfTeX-1.40.17, and has been sent on pdf-archive.com on 07/09/2017 at 19:09, from IP address 110.22.x.x. The current document download page has been viewed 195 times.
File size: 105.33 KB (2 pages).
Privacy: public file










File preview


MTH3241

Lecture 1

Randomness

A monkey randomly types letters fA; B; C; D; E; F; G; H; I; J; K; L; M; N; O; P; Q; R; S; T; U; V; W; X; Y; Zg. What
is the expected number of key presses until ABRACADABRA is generated?
Solution
The transition matrix for this markov chain is given by
0
25
1
0
0
0
0
26
26
B 24
1
1
B 26 26
0
0
0
26
B
1
B 24 1
0
0
0
B 26 26
26
B 25
1
0
0 26
0
B 26 0
B 23 1
1
1
B
0
0
26
B 26 26 26
B 25 0
0
0
0
0
B
P D B 26
1
1
B 23
0
0
0
B 26 26 26
B 25 0
0
0
0
0
B 26
B 24 1
0
0
0
B 26 26 0
B 24 1
B
0
0
0
0
B 26 26
B 25 0
0
0
0
0
@ 26
0
0
0
0
0
0

0
0
0
0
0

0
0
0
0
0
0

1
26

0
0
0
0
0
0

0
0
0
0
0
0
0

1
26

0
0
0
0
0
0
0
0

1
26

0
0
0
0
0

0
0
0
0
0
0
0
0
0

1
26

0
0
0
0

1
26

0
0
0

0
0

A diagram for this system can be seen in the following figure.

¿
abracadabr

abracadab

a

a

r

a

b

r

b

ab

abracada

a

abracad

abr

abraca

d

abra
abrac

a

The absorption equations are described by
i D 1 C

m
X

pij j :

j D1

This leads to a system of linear equations in 1 ;    ; 12 .

1 of 2

c

a

0
0
0
0
0
0
0
0
0
0
1
26

1

1
C
C
C
C
C
C
C
C
C
C
C
C
C:
C
C
C
C
C
C
C
C
C
C
A

MTH3241

Lecture 1

Randomness

8
1
ˆ
C 2  26
1 D 1 C 1  25
ˆ
26
ˆ
ˆ
ˆ
1
1
ˆ
2 D 1 C 1  24
C 2  26
C 3  26
ˆ
26
ˆ
ˆ
24
1
1
ˆ
ˆ
3 D 1 C 1  26 C 2  26 C 4  26
ˆ
ˆ
ˆ
1
ˆ
4 D 1 C 1  25
C 5  26
ˆ
ˆ
26
ˆ
ˆ
23
1
1
ˆ
5 D 1 C 1  26 C 2  26
C 3  26
C 6 
ˆ
ˆ
ˆ
< D 1 C   25 C   1
7 26
6
1 26
1
1
ˆ
7 D 1 C 1  23
C 2  26
C 3  26
C 8 
ˆ
26
ˆ
ˆ
25
1
ˆ
ˆ

D
1
C


C


8
1 26
9 26
ˆ
ˆ
ˆ
24
1
1
ˆ

D
1
C


C

ˆ
9
1
2  26 C 10  26
ˆ
26
ˆ
ˆ
24
1
1
ˆ
10 D 1 C 1  26 C 2  26 C 11  26
ˆ
ˆ
ˆ
ˆ
ˆ11 D 1 C 1  25 C 12  1
ˆ
26
26
ˆ
ˆ
: D 0
12
Represented in matrix form:
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@

1
26
24
26
24
26
25
26
23
26
25
26
23
26
25
26
24
26
24
26
25
26

0

1
26
25
26
1
26

0
1
26

0

1
0

1
26

1
26

0

0

1
26

1
26

0

0
0
0
0
0

1
26
1
26

0
0

This matrix row Reduces to
0
1 0 0
B
B0 1 0
B
B0 0 1
B
B0 0 0
B
B0 0 0
B
B0 0 0
B
B0 0 0
B
B0 0 0
B
B0 0 0
B
B0 0 0
B
@0 0 0
0 0 0

0
0

0
0
0
1
0
0
0
0
0
0
0
0

0
0
0

1
26

1
0
0
0
0
0
0
0
0

0
0
0
0
1
0
0
0
0
0
0
0

0
0
0
0
0
1
0
0
0
0
0
0

1
26

1
0
0
0
0
0
0
0

0
0
0
0
0
0
1
0
0
0
0
0

0
0
0
0

0
0
0
0
0

1
26

1
26

1
0
0
0
0
0
0

0
0
0
0
0
0
0
1
0
0
0
0

0
0
0
0
0
0

1
0
0
0
0
0

0
0
0
0
0
0
0
0
1
0
0
0

0
0
0
0
0
0
0
0
0
1
0
0

0
0
0
0
0
0
0

1
26

1
26

1
0
0
0
0

0
0
0
0
0
0
0
0
0
0
1
0

1
0
0
0

0
0
0
0
0
0
0
0
0
0
0
1

0
0
0
0
0
0
0
0
1
26

1
0
0

1
26
1
26

0
0
0
0
0
0
0
0
0
1
26

1
0

0
0
0
0
0
0
0
0
0
0
1
26

1

3670344487444778
3670344487444752
3670344487444102
3670344487427202
3670344486987776
3670344475563402
3670344178528976
3670336455634602
3670135660380176
3664914983765126
3529177391773826
0

Therefore, if one key is pressed per second, the expected time to ABRACADABRA is
 116; 306; 000 years:

2 of 2

1
1
1
1
1
1
1
1
1
1
1
0

1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A

1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A






Download Lecture01rnd



Lecture01rnd.pdf (PDF, 105.33 KB)


Download PDF







Share this file on social networks



     





Link to this page



Permanent link

Use the permanent link to the download page to share your document on Facebook, Twitter, LinkedIn, or directly with a contact by e-Mail, Messenger, Whatsapp, Line..




Short link

Use the short link to share your document on Twitter or by text message (SMS)




HTML Code

Copy the following HTML code to share your document on a Website or Blog




QR Code to this page


QR Code link to PDF file Lecture01rnd.pdf






This file has been shared publicly by a user of PDF Archive.
Document ID: 0000669853.
Report illicit content