1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of...
-
Upload
emmeline-simon -
Category
Documents
-
view
214 -
download
2
Transcript of 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of...
![Page 1: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/1.jpg)
1
Introduction to Stochastic Processes and Markov Chain
Prof. Dr. Md. Asaduzzaman Shah Department of Statistics
University of Rajshahi
![Page 2: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/2.jpg)
2
Variable
Values varies from individual to individual
![Page 3: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/3.jpg)
Stochastic Variable
Families of random variable
3
![Page 4: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/4.jpg)
One dimensional process
Classified into four types of processes
(i) Discrete time, discrete state space (ii) Discrete time, continuous state space (iii) Continuous time, discrete state space (iv) Continuous time, continuous state space
4
![Page 5: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/5.jpg)
Discrete time Discrete state space
5
![Page 6: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/6.jpg)
Examples
A fair coin is tossed n times and the outcomes represented by Xn. Here, Toss No. treated as ‘time’ and the result/outcome considered as ‘space’ of the experiment. Both time and space are discrete.
Toss No. 1 2 3 … n-1 n
Outcomes T H H … H T
6
![Page 7: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/7.jpg)
Experiment and Its Outcomes
Suppose a fair coin is tossed twice. Sample space and four simple events are
4321 ,,,,,, TTTHHTHH
TTandTHHTHH 4321 ,,
7
![Page 8: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/8.jpg)
Counting Heads
0)()(
1)()(
1)()(
2)()(
4
3
2
1
TTXX
THXX
HTXX
HHXX
8
![Page 9: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/9.jpg)
Probability
Now the values 0,1 and 2 have probabilities ¼, 2/4=1/2 and ¼. So, X(t) = X(ω) = 0,1,2 are stochastic variable with t=0,1,2.
9
![Page 10: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/10.jpg)
Discrete time Continuous state space (Velocity of a car in a time interval (0,t)
10
![Page 11: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/11.jpg)
DC
In a stochastic process {X(t), t€T} in which time parameter, t is discrete, however, the state space, velocity X(t), which is increasing or decreasing is continuous.
Time(Road) 10:00:00 12:30:00 18:45:00 24:00:00
Velocity 60m/h 45m/h 26m/h 70m/h
11
![Page 12: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/12.jpg)
Continuous time, discrete state space
12
![Page 13: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/13.jpg)
CD
The number of passengers waiting at Dhaka Bus Terminal, Rajshahi on Friday, August 09, 2011 from (10-11 AM).Time and tide wait for none(Continuous).
Time 10:00 10:15 10:20 11:00
Passengers
waiting460 345 220 370
13
![Page 14: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/14.jpg)
More examples
Long Route Buses: Hanif, National , Green Line, etc. that’s are started in a particular time.
Patients are admitted at Rajshahi Medical College Hospital (RMCH) in a particular day.
Customer’s are waiting in a super-market for getting Eid Fashion
14
![Page 15: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/15.jpg)
Continuous time Continuous state space
15
![Page 16: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/16.jpg)
CC
If we measure the temperature of a certain place in a time interval (0,t) for each day is an example of CTCSS, because time interval is continuous as well as temperature which is not fixed, it is rising/falling following a day follows morning to noon, noon to evening, evening to night, so state space is also continuous.
16
![Page 17: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/17.jpg)
Loss variables character
** Making Laugh to see the teeth (^^^^^^) instead of monitoring the defective teeth
** Slapping to see the number of eye drops (weeping),
17
![Page 18: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/18.jpg)
Notations
Discrete random variable represented by Xn or X(n).
Continuous random variable indicated by Xt or X(t).
Stochastic processes are defined by the notations { Xn} and {X(t)}
18
![Page 19: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/19.jpg)
Differences
Discrete Continuous
Family represented
by {Xn, n=0,1,2,…,} Family represented by {Xt, t€T} or
{X(t), t€T}, T is finite/infinite.
Parameter ‘n’ Parameter ‘t’ interpreted as time, some times it represents characters like as, (i) distance, (ii) length, (iii) thickness and so on.
Stochastic sequence Stochastic process
19
![Page 20: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/20.jpg)
At a Glance
Xn
X(t)
Xn
Time
X(t)
Thrown/Toss No.
Distance/Length/
Thickness.
Outcomes of ExperimentOr Space
20
![Page 21: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/21.jpg)
Relationship
In some of the cases, the r.v Xn, i.e. members of the family {Xn, n>=0} are mutually independent, but more often they are not independent. We generally come across Processes whose members are mutually dependent.
Throwing a fair coin 20 times by 20 peoples, outcomes will be different instead of, a person will throw the coin 20 times. The relationship among them is often of great importance.
The nature of dependence could be infinitely varied. Dependence of such special types, which occurs quite often and is
of great importance. According to the nature of dependence relationship existing among
the members of the family, we may describe or mathematically formulate them by some Stochastic Processes.
21
![Page 22: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/22.jpg)
Stationary Processesstochastic process { Xn, n>=1 } is time independent (not necessary hours, months, years) i.e. covariance stationary.
Let Xn, n>=1 be uncorrelated random variables with mean 0 and variance 1.
nmifnmif
mn
mnmnmn
XXE
XEXEXXEXXmnC
01
),(
)(.)(),(,cov,
22
![Page 23: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/23.jpg)
1. Poisson Process
Consider the process {X(t), t€T} with
;...;3,2,1,0,0,!
)()(Pr
nn
tentX
nt
23
![Page 24: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/24.jpg)
(i) the mean function and (ii) the variance
ttXEtm )()(
ttX )(var
24
![Page 25: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/25.jpg)
Non-Stationary
Both are functions of t. Therefore, the stochastic process {Xn, n>=1} is non-stationary/evolutionary. The distribution (Poisson) of the stochastic process {Xn, n>=1} is functionally dependent on t.
25
![Page 26: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/26.jpg)
2. Mean and Co-variance
tddtDDEtXEtm 2121)()(
222
212
11
22
2221
21
21
2221
21
2121
))(()()(
)())(()(
)()(
)()()()(
))(()()(
DEDVartimesofproduct
ddtimesofsumDEDVar
dstddtsd
DEstDDEtsDE
sDDtDDEsXtXE
26
![Page 27: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/27.jpg)
Variance
)(2
))(()(var)(
))(()()(var
22
22
221
21
21
221
22
221
22
22
dtdtddtddt
tXEtXtXE
tXEtXEtX
27
![Page 28: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/28.jpg)
The Process is non-stationary, i.e. evolutionary
22
21)(),(cov),( tssXtXtsC
28
![Page 29: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/29.jpg)
3. Expection
2210)( tDtDDtX
0.00.0)()()()]([ 22
210 ttDEtDEtDEtXE
29
![Page 30: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/30.jpg)
Covariance
22
2210
2210
2210
2210
1
)()()})({(
))(()).(()}()({)(),(
stts
sDsDDEtDtDDEsDsDDtDtDDE
sXEtXEsXtXEsXtXCov
30
![Page 31: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/31.jpg)
Variance
422
41
20
2210 1)()()()var{)}({ ttDvtDvtDvtDtDDtXVar
31
![Page 32: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/32.jpg)
Mean stationary but Process non-stationary
Therefore, the stochastic process, X(t)=D0 + D1 t + D2 t2, where Di, i=0,1,2 are
uncorrelated random variables with mean 0 and variance 1, is dependent on the time parameter t and s, so it is non-stationary. But the mean of the process is stationary.
32
![Page 33: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/33.jpg)
Briefly: Time, also treated as distance, length, thickness etc.
StationaryTime- Independent
Non-Stationary
Time- Dependent
33
![Page 34: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/34.jpg)
Problem
Manufacturers A and B are competing with each other in a restricted market. Over the year, A’s customers have exhibited a high degree of loyalty as measured by the fact that customers using A’s product 80 percent of the time. Also former customers purchasing the product from B have switched back to A’s 60 percent of the time.
(a) Construct and interpret the state transition matrix terms of (i) retention and loss, (ii) retention and gain.
(b) Calculate the probability of a customer purchasing A’s product at the end of the second period, Draw the transition probability diagrams and the transition trees.
34
![Page 35: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/35.jpg)
State Transition P
Current Purchase
(n=0)
Next Purchase (n=1)
A B
A 0.8 0.2
B 0.6 0.4
35
![Page 36: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/36.jpg)
Interpretation of Conditional Probability
The probability of a customer’s purchase at the next step (n=1) depends upon the product that he bought previously (n=0), i.e., current purchase.
36
![Page 37: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/37.jpg)
Retention and Loss (Rows of P),
P11= 0.80
means that a customer now using A’s product will again purchase A’s product at the next purchase in 8 of 10 times. This implies retention to A’s product.
P12= 0.20
means that the customer now using A’s product will switch over to B’s product at the next purchase in 2 pot of 10 times. This implies loss of A’s product.
37
![Page 38: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/38.jpg)
Retention and Gain (Columns of P),
P11= 0.80
means that the customer now using A’s product will again purchase A’s product next time in 8 out of 10 times. This implies retention of A’s product.
P12= 0.60
shows that the customer now using B’s product will purchase A’s product next time in 6 out of 10 times. This implies gain to A’s product.
38
![Page 39: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/39.jpg)
Second Row and Column
A similar interpretation holds for the second row of P
The second column of P can be interpreted similarly.
39
![Page 40: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/40.jpg)
Transition Probability Diagram
0.80 A B 0.40
0. 200.
60
40
![Page 41: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/41.jpg)
Transition Trees
A
A
B
A
A
B
B
0.80
0.20
0.80
0.20
0.40
0.60
41
![Page 42: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/42.jpg)
QUESTIONNAIRE ON PROFESSIONS (Research Project & Field Studies)
•AgricultureBusinessServiceOthers
42
![Page 43: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/43.jpg)
Generations and their professions
(Please put tick () mark on the Boxes)
Agriculture
Service
Others
Business
Great Grand FatherGrand Father
Father
43
![Page 44: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/44.jpg)
Raw data table
ProfessionsGreat Grand Father Grand Father Father
A B S O A B S O A B S O
X X X X X X X X X
X X X X X X X X X
X X X X X X X X X X X X X X X X X X
X X X X X X X X X
44
![Page 45: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/45.jpg)
NUMERICAL FIGURES IN DIFFERENT STATES OF CLASSES
20062421149Total
165434O
60501S
1810116B
1600157138A
OSBA
TotalG G Father
G Father
Pro
fess
ions
Professions
45
![Page 46: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/46.jpg)
ESTIMATED TRANSITION PROBABILITY
A B S O
A 0.8625 0.0437 0.0937 0.0000
B 0.3333 0.6111 0.0000 0.0555
S 0.1666 0.0000 0.8333 0.0000
O 0.2500 0.1875 0.2500 0.3125
G G Father
G Father
Pro
fess
ions
Professions
46
![Page 47: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/47.jpg)
EXPECTED STAY IN EACH SOCIAL CLASS AND PRAISE MEASURE OF SOCIAL MOBILITY
1.4904 1.0064 1.4999 Others
1.82552.73905.0000 Service
2.2702 1.21152.7503 Business
2.2854 1.2253 2.8003 Agriculture
Class )1(
1)(
ijj p
E
)1(
1)(
jj p
E
)(
)(
j
jj E
E
47
![Page 48: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/48.jpg)
COMMENT
Last column of the above Table states that the generation of Great Grand Father of Grand Father, the agriculture community has most tendency to adjust their children to agriculture. Then comes the Business community and third after Agriculture and Business community comes the Service community. Finally comes the others community.
48
![Page 49: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/49.jpg)
Bartholomew Model
k
j
k
iiji jipf
1 1
49
![Page 50: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/50.jpg)
This measure depends on which generation we choose as our base. Replace fi by pj we get
measure of social mobility given by
10,1 1
DjippDk
j
k
iijj
50
![Page 51: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/51.jpg)
Conditions
D = 0,
the society is perfectly stable i.e, no mobility takes place
D = 1,
the society is perfectly mobile.
51
![Page 52: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/52.jpg)
Application Bartholomew Method
D = p1 [ p12+ 2p13 + 3p14 ] + p2 [ p21+ p23
+ 2p24 ] +
p3 [ 2p31+ p32 + p34 ] + p4 [ 3p41+ 2p42
+ p43 ]
= 0.2242
From the Bartholomew co-efficient of mobility we see that the society of that time was quite stable one.Indicates that the society in the generation of Great Father to Grand Father had a very good deg of mobility which is an indicator of the economic and social progress the society is going through in this generation.
52
![Page 53: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/53.jpg)
Chi-Square Test
Simplest and most commonly used non-parametric test in statistical work. The quantity describes the magnitude of discrepancy theory and observation.
53
![Page 54: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/54.jpg)
Hypothesis
The transition probability matrix (t.p.m) in internal is completely independent of the lithology at the immediate underlying point. Under this hypothesis the expected t.p.m would consist of rows that are all identical to the fixed probability vector.
54
![Page 55: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/55.jpg)
Theory
ectedEobservedO
E
EO
j j
jj
exp,
,)(4
1
22
55
![Page 56: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/56.jpg)
Values
)(48.9
)(39.142
05.0,4
2
talculated
calculated
56
![Page 57: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/57.jpg)
Decision
So the null hypothesis is rejected as the tabulated value is lower than the calculated value.
57
![Page 58: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/58.jpg)
NUMERICAL FIGURES IN DIFFERENT STATES OF CLASSES
TotalA B S O
A 75 31 40 3 149
B 0 15 6 0 21
S 5 3 15 1 24
O 3 0 1 2 6
Total 149 21 24 6 200
Professions
Pro
fess
ions
Father
G Father
58
![Page 59: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/59.jpg)
EXPECTED STAY IN EACH SOCIAL CLASS AND PRAISE MEASURE OF SOCIAL MOBILITY
Class
Agriculture 1.8923
Business 1.2845
Service 2.3233
Others 1.3529
)(
)(
j
jj E
E
59
![Page 60: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/60.jpg)
COMMENT
From the above Table, the generation from Grand Father to Father, the service community has most tendency to adjust their son to service. Second comes to agriculture community and third comes the business.
60
![Page 61: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/61.jpg)
D-index & Chi-square
Using the measure of mobility as given by Bartholomew, D = 0.6031, A good degree of mobility, indicates fast social and economic progress.
Null hypothesis accepted, generations have independence in profession selections
61
![Page 62: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/62.jpg)
Occupational Mobility : Markov Approach
Human Societies Stratified
Income Occupation Social Status
Residence Place
62
![Page 63: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/63.jpg)
societies move on class
• In our society children do not always follow their fathers’ footsteps.
• In a free society a person has some degree of choice about changing his job or moving house.
• The inherent uncertainty of individual behavior in these situations means that the future development of the mobility process cannot be predicted with certainty but only in terms of probability.
63
![Page 64: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/64.jpg)
MODELS FOR SOCIAL MOBILITY
• A very simple model for the development of a single family line and then, investigate the consequences of unrealistic features. The fundamental requirement in a model is that it must specify the way in which changes in social occur.
64
![Page 65: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/65.jpg)
Assumption of model• Chance of moving depends only on the
• present class • but not on the
• past class/remote past• If the movement can be regarded as taking place at
discrete points in time the appropriate model becomes a simple
• Markov chain• Changes are governed by transition probabilities
which are independent of time. 65
![Page 66: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/66.jpg)
Markov
• Andrei Andreivich Markov
• Alive 67 years
• From1856 to 1922
• Russian Mathematician
• Idea first occurred to him when he was watching an opera of the famous Russian writer Pushkin’s.
66
![Page 67: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/67.jpg)
Notations
• Pij, the probability that the son of a father in class i is in class j (since the system is closed)
• 1
1
k
jijP
k is the number of classes and P, the matrix of transition probabilities.
TPpTP )0()(
67
![Page 68: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/68.jpg)
Day-to-Day Inventory
Problem : The number of units of an item that are withdrawn from inventory on a day-to-day basis is a Markov chain in which requirements for tomorrow depend on today’s requirements. A one day transition matrix is given below :
68
![Page 69: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/69.jpg)
Number Units Withdrawn from Inventory
6.03.01.012
4.03.03.010
0.04.06.05
12105
Today
Tomorrow
69
![Page 70: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/70.jpg)
Target
• (a) Tree diagram showing inventory requirements on two consecutive days.
• (b) A two-day transition matrix
• (c) How a two-day transition matrix help manager for inventory management.
70
![Page 71: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/71.jpg)
Transition Tree
5 10
5
12
5
10
12
5
10
12
10
12
5
0.6
0.4
0.0
0.6
0.3
0.1
0.4
0.3
0.3
0.0
0.4
0.6
71
![Page 72: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/72.jpg)
Transition Probabilities
48.0)1.0)(0.0()3.0)(4.0()6.0)(6.0(211 p
36.0)3.0)(0.0()3.0)(4.0()4.0)(6.0(212 p
16.0)6.0)(0.0()4.0)(4.0()0.0)(6.0(213 p
48.0)1.0)(0.0()3.0)(4.0()6.0)(6.0(211 p
72
![Page 73: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/73.jpg)
Transition Probabilities
31.0)1.0)(4.0()3.0)(3.0()6.0)(3.0(221 p
33.0)3.0)(4.0()3.0)(3.0()4.0)(3.0(222 p
36.0)6.0)(4.0()4.0)(3.0()0.0)(3.0(223 p
73
![Page 74: 1 Introduction to Stochastic Processes and Markov Chain Prof. Dr. Md. Asaduzzaman Shah Department of Statistics University of Rajshahi.](https://reader035.fdocuments.us/reader035/viewer/2022070412/5697bf821a28abf838c858f8/html5/thumbnails/74.jpg)
Transition Probabilities
21.0)1.0)(6.0()3.0)(3.0()6.0)(1.0(231 p
31.0)3.0)(6.0()3.0)(3.0()4.0)(1.0(232 p
48.0)6.0)(6.0()4.0)(3.0()0.0)(3.0(233 p
74