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