Download STA 348 Introduction to Stochastic Processes Lecture 1

STA 348
Introduction to
Stochastic Processes
Lecture 1
1
Adminis-trivia

Instructor: Sotirios Damouras


Contact Info:



Pronounced Sho-tee-ree-os or Sam
email: [email protected]
Office hours: SE/DV 4062, every Mon 2-4pm and
Tues 4-5pm, or by appointment (email)
Course web page:


https://portal.utoronto.ca/ (UofT Portal)
All course material (outline, lecture slides,
assignments & solutions) posted on portal
Outline

Textbook:



Introduction to probability models 10th Ed, by Sheldon
M. Ross (in bookstore)
Cover (parts of) § 1-8, & extra topics if time permits
Evaluation:



9 Weekly Assignments, best 8/9 worth 20%
 Due @ start of tutorial, NO late submissions
2 Term Tests, worth 20% each
 NO make-up tests. Weight shifted to Final exam with
UofT Medical note AND absence declaration on ROSI
Final Exam, worth 40%
Important Dates
Sept
Oct
Nov
LEC (TUE 2-4pm
@ IB 220)
LEC (THU 3-4pm
@ IB 200)
TUT (FRI 3-4pm
@ IB 200)
6
8
9 no tutorial
13
15
16 assign 1 due
20
22
23 assign 2 due
27
29
30 assign 3 due
4
6
7 assign 4 due
11 Midterm 1
13
14 no tutorial
18
20
21 assign 5 due
25
27
28 assign 6 due
1
3
4 assign 7 due
8 Midterm 2
10
11 no tutorial
15
17
18 assign 8 due
22
24
25 assign 9 due
4
What Is This Course About?

Modeling & analyzing behavior of a collection
of dependent random variables (RV’s)


(X1,X2,…) = {Xt}t=1,2,… is a Stochastic Process
How is this different from Statistics?


Statistics: X1, X2, … is independent random
sample from some distribution/population
Stochastic Processes: { Xt }t=1,2,… is collection of
dependent RV’s, describing a random process at
different points t=1, 2,… in time or space

E.g. Country’s population at year t = 1, 2, …
5
Example 1

Gamble $10 in Roulette (betting on
red / black) till you double or loose it



If you win bet on red/black, you double bet amount
P(winning individual bet) = 18/(36+2) = .4737
Which is the best strategy for maximizing the
chance of doubling money (reaching $20):
A.
B.
C.
Bet $10 all at once
Bet $1 at a time
It doesn’t matter
6
Example 2

You are tossing a fair coin, i.e. P(Heads) =
P(Tails) = ½, and counting the # of tosses till
one of two patterns occurs


Pattern 1 = (H,H) & Pattern 2 = (H,T)
Which pattern appears first on average?
A.
B.
C.
Pattern 1
Pattern 2
Both are equally likely
7
Example 3

A type of bacterium reproduces in
the following way:



With prob. ½ it splits into 2 identical copies
With prob. ½ it dies before dividing
If you place 10 such bacteria on a Petri dish,
what happens to their (long-run) population
A.
B.
C.
It will certainly survive indefinitely
It will certainly die out eventually
It will can either survive or die (w/ some prob’s)
8
Example 4

Consider service queue (e.g. airport security)



People arrive at rate λ, and
People get served at rate μ (λ < μ)
If rate λ doubles, how should rate μ change
so that the mean time a person stays in the
system (wait + service time) stays the same?
A.
B.
C.
μ should double
μ should less than double
μ should more than double
9
Stochastic Processes

How to analyze collection of RV’s?


If RV’s are independent work with marginals


f1,2,...  x1 , x2 ,...  f1  x1   f 2  x2   ... (as in Stats)
If RV’s are dependent, work with conditionals


E.g. {Xt}t=1,2,… with joint pdf f1,2,...  x1 , x2 ,...
f1,2,...  x1 , x2 ,...  f1|2,...  x1 | x2 ,...  f 2|3,...  x2 | x3 ,...  ...
Stochastic Processes mostly deal with
various “types” of conditional dependence

But first, need to brush up our Probability Theory
10
Experiments & Events




Experiment: process with random result
Outcome: elementary result of experiment
Sample Space (S): Set of all outcomes
Event: An arbitrary collection of outcomes


Events are subsets of S, denoted by capital letters
E.g. Rolling a 6-sided die, E = {even roll} = {2,4,6}
Venn Diagram:
outcomes
●1
●3
●5
●2 ●4
●6
S
event E
11
Combining Events



Union:

A  B  {A or B}

in1 Ai  A1  A2 
A
 An
Intersection:
A

A  B  A  B  {A and B}

in1 Ai  A1  A2 
Complement:

AUB
B
A∩B
B
 An
A
AC
Ac  {not A}
12
De Morgan’s Laws
 A  B



C
c
c
A

B

A

B


c
 A  B
c
A B
 A  B
 Ac  Bc
C
A B
More generally:
n
i 1
Ai   A1c  A2c 
 Anc

n
i 1
Ai   A1c  A2c 
 Anc
c
c
13
Probabilities

Consider an experiment with sample space S.
A probability (measure) is a function P(·) that
assigns numbers P(A) to events A⊂S, so that:
1. P  A   0
2. P  S   1
3. If A1 , A2 , A3 ,
then P 
Events A1,A2 ,

i 1
are mutually exclusive events,
Ai    i 1 P  Ai 

are mutually exclusive if Ai  Aj   for all i  j
14
Conditional Probability &
Independence

Conditional Probability: P(A|B) is probability of
event A given that event B has occurred
P  A  B
P  A | B 
, for P  B   0
P  B

Independence: Events A, B are independent if

 P  A | B   P( A)
P  A  B   P  A P  B   

 P  B | A  P( B)
15
Mutual Independence
Generalization to n≥2 events:

A finite collection of events  A1 , A2 ,
, An  is called
(mutually ) independent if for any sub-collection
A , A
k1

k2
,

, Akm : P

m
i 1

 
Aki  i 1 P Aki
m
Pairwise indep. does not imply mutual indep.
P  Ai  Aj   P  Ai  P  Aj  , i  j 
  A1 , A2 ,
, An  are mutually independent
16
Example

S=
(H,H) (H,T)
(T,H) (T,T)
Consider flipping two fair coins & define events
A={(H,H),(H,T)}, B={(H,H),(T,H)}, C={(H,H),(T,T)}

Are A, B, and C pairwise independent?

Are A, B, and C mutually independent?
17
Rules of Probability

c
P
(
A
)  1  P( A)
Complement Rule:

Addition Rule:
P( A  B)  P( A)  P( B)  P( A  B)


B
A
If A, B mutually exclusive,
then P( A  B)  P( A)  P( B)
A B
Multiplication Rule:
P( A  B)  P( A | B) P( B)  P( B | A) P( A)

If A, B independent,
then P( A  B)  P( A) P( B)
18
Rules of Probability

Generalizations for n≥2 events: For any finite
collection of events {A1,A2,...,An}

Addition Rule:
P

n
i 1

Ai   i 1 P  Ai    i  j P  Ai  Aj  
n
  i  j k P  Ai  Aj  Ak  

  1
n 1
P

n
i 1
Ai
Multiplication Rule:
P


n
i 1

Ai  P  A1   P  A2 | A1   P  A3 | A1  A2  
 P  An | A1  A2 
 An 1 
19

Law of Total Probability

S
Partition of S is finite set of
B2
events {B1,B2,...,Bn}, such that:
Bi  B j  , i  j &

n
i 1
Bi  S
B1
A
B3
For any event A and partition {B1,B2,...,Bn},
P  A  i 1 P  A  Bi   i 1 P  Bi  P  A | Bi 
n

From addition
rule, since:
n

 A   A  B1    A  B2     A  Bn 


and  A  Bi    A  B j   , i  j 20
Bayes’ Formula

Let {B1,B2,...,Bn} be a partition of S such that
P(Bi)>0, for i=1,2,...,n. Then, for any event A
P  B j | A 


P  Bj  P  A | Bj 

n
i 1
P  Bi  P  A | Bi 
P( B) P( A | B)
For n=2: P  B | A 
P( B) P( A | B)  P( B c ) P( A | B c )
Method for revising event Bj’s probability, given
information on occurrence of another event A


Know: P(Bj) prior probability, P(A|Bi), i=1,…,n
Want: P(Bj|A) posterior probability
21
Counting Rules

Permutation Rule: Number of permutations of r
objects, selected w/o repeats from n objects:
Prn  n  (n  1) 

 (n  r ) 
n!
, where 0  r  n
 n  r !
Combination Rule: Number of combinations of r
objects, selected w/o repeats from n objects:
n
n


P
n!
Crn     r 
, where 0  r  n
 r  r ! r ! n  r !

Binomial Theorem:
 x  y
n
 n  i n i
  i 0    x  y
i
n
22
Example

(Matching Problem: §1-Q32) At a party, #n
people get drunk & on their way out they grab
a coat at random. What is the probability that
nobody got their own coat?
23
24
Example

n# points are randomly drawn on a circle.
What is the probability that all points lie in a
semi-circle?
25
26