Introduction
The gambler’s ruin
Applications of random walks
Introduction to random variables and an
example of random walk
PhD Away Days 2017
Notarnicola Massimo
September 29, 2017
Notarnicola Massimo PhD Away Days
Introduction
The gambler’s ruin
Applications of random walks
1 Introduction
Examples of random events
Random variable
Probability measure
2 The gambler’s ruin
The problem
Solution
3 Applications of random walks
Notarnicola Massimo PhD Away Days
Introduction
The gambler’s ruin
Applications of random walks
Examples of random events
Random variable
Probability measure
1 Introduction
Examples of random events
Random variable
Probability measure
2 The gambler’s ruin
The problem
Solution
3 Applications of random walks
Notarnicola Massimo PhD Away Days
Introduction
The gambler’s ruin
Applications of random walks
Examples of random events
Random variable
Probability measure
1 Coin tossingpossible outcomes: ⌦ = {H (head) , T (tail) }
2 Dice rollingpossible outcomes: ⌦ = {1, 2, 3, 4, 5, 6}
To any possible outcome we associate a value ! random variable
Notarnicola Massimo PhD Away Days
Introduction
The gambler’s ruin
Applications of random walks
Examples of random events
Random variable
Probability measure
A random variable is a function
X : ⌦ ! E,! 7! X(!) ,
where
• ⌦ is the set of possible outcomes,
• E is a set (e.g. E = R)
Notarnicola Massimo PhD Away Days
Introduction
The gambler’s ruin
Applications of random walks
Examples of random events
Random variable
Probability measure
Back to examples:
1 Coin tossingDefine X : {H,T} ! {0, 1}
X(H) = 0, X(T ) = 1
2 Dice rollingDefine X : {1, . . . , 6} ! {1, . . . , 6}
X(k) = k , 8k = 1, . . . , 6
assign probabilities to random events ! probability distribution
Notarnicola Massimo PhD Away Days
Introduction
The gambler’s ruin
Applications of random walks
Examples of random events
Random variable
Probability measure
Let X : ⌦ ! E be a random variable.
The probability distribution of X is the map
PX
: P(E) ! [0, 1],A 7! P
X
(A) := P(X(!) 2 A) ,
given by
P(X(!) 2 A) =
Z
⌦{!:X(!)2A}dP(!)
=X
x2AP(X(!) 2 {x}) .
Notarnicola Massimo PhD Away Days
Introduction
The gambler’s ruin
Applications of random walks
Examples of random events
Random variable
Probability measure
The map PX
is a probability measure, i.e. satisfies
• PX
(⌦) = 1,PX
(;) = 0
•for any countable collection {A
i
}i2I of disjoint sets,
PX
([
i2IA
i
) =X
i2IPX
(Ai
)
Notarnicola Massimo PhD Away Days
Introduction
The gambler’s ruin
Applications of random walks
The problem
Solution
1 Introduction
Examples of random events
Random variable
Probability measure
2 The gambler’s ruin
The problem
Solution
3 Applications of random walks
Notarnicola Massimo PhD Away Days
Introduction
The gambler’s ruin
Applications of random walks
The problem
Solution
A gambler starts with initial fortune of k euros. In each game, he
•wins 1 euro with probability p
•loses 1 euro with probability q = 1� p
Objective: reach a total fortune of K > k euros before running
out of money (ruin).
The game stops either when he reaches K euros or when he is
ruined.
Notarnicola Massimo PhD Away Days
Introduction
The gambler’s ruin
Applications of random walks
The problem
Solution
• Xn
2 {�1, 1}, n � 1 denote the gain of the n-th game,
P(Xn
= 1) = p ,P(Xn
= �1) = q = 1� p
• Sn
, n � 0 denote the gambler’s fortune after the n-th game,
S0 = k ,
Sn
= k +X1 +X2 + . . .+Xn
, n � 1
• ⌧k
the time when the game stops,
⌧k
:= min{n � 0 : Sn
= 0 or Sn
= K|S0 = k}
Question: What is the probability that the gambler is ruined
pk
:= P(S⌧k = 0) ?
Notarnicola Massimo PhD Away Days
Introduction
The gambler’s ruin
Applications of random walks
The problem
Solution
k
K
⌧k
n
Sn
Figure: Random walk
Notarnicola Massimo PhD Away Days
Introduction
The gambler’s ruin
Applications of random walks
The problem
Solution
We have
p0 = P(S⌧0 = 0) = 1
pK
= P(S⌧K = 0) = 0
Computation of pk
= P(S⌧k = 0):
k k + 1k � 1
pq
pk
= pk�1 ⇥ q + p
k+1 ⇥ p
Notarnicola Massimo PhD Away Days
Introduction
The gambler’s ruin
Applications of random walks
The problem
Solution
We obtain a recurrence relation of order 2,
ppk+1 � p
k
+ qpk�1 = 0 , 0 < k < K ,
p0 = 1, pK
= 0 .
Associated characteristic equation: px2 � px+ q = 0Discriminant: (2p� 1)2
! distinguish cases p = 1/2 and p 6= 1/2
Notarnicola Massimo PhD Away Days
Introduction
The gambler’s ruin
Applications of random walks
The problem
Solution
Rec. relation:
⇢pp
k+1 � pk
+ qpk�1 = 0 , 0 < k < K
p0 = 1, pK
= 0
Char. equation: px2 � px+ q = 0
p = 1/2 p 6= 1/2
x = 1 x = 1±|2p�1|2p 2 {1, q/p}
pk
= a+ bk pk
= a+ b(q/p)k
pk
= 1� k/K pk
= (q/p)k�(q/p)K
1�(q/p)K
Notarnicola Massimo PhD Away Days
Introduction
The gambler’s ruin
Applications of random walks
1 Introduction
Examples of random events
Random variable
Probability measure
2 The gambler’s ruin
The problem
Solution
3 Applications of random walks
Notarnicola Massimo PhD Away Days
Introduction
The gambler’s ruin
Applications of random walks
! random walks describe a path consisting of a succession of
random steps
Some applications of random walks:
•finance: stock market prices
•physics: random movement of molecules in a liquid (Brownian
motion)
•etc.
Notarnicola Massimo PhD Away Days
Introduction
The gambler’s ruin
Applications of random walks
Figure: Simulation of a Brownian motion
Notarnicola Massimo PhD Away Days
Introduction
The gambler’s ruin
Applications of random walks
Thank you for your attention!
Notarnicola Massimo PhD Away Days