Introduction to Probability: Lecture 22: The Poisson ... · The Poisson process • Readings: Start Section 5.2. Lecture outline • Review of Bernoulli process • Deﬁnition of

LECTURE 22: The Poisson process

• Definition of the Poisson process

- applications

• Distribution of number of arrivals

• The time of the kth arrival

• Memorylessness

• Distribution of interarrival times

Applications of the Poisson process

(1781-1840)

• Deaths from horse kicks in the Priussian army (1898)

• Placement of phone calls, service requests, etc.

• Particle emissions

• Photon arrivals from a weak source

The Poisson PMF for the number of arrivals

LECTURE 16

The Poisson process

• Readings: Start Section 5.2.

Lecture outline

• Review of Bernoulli process

• Definition of Poisson process



• Other properties of the Poisson process

Bernoulli review

• Discrete time; success probability p

• Number of arrivals in n time slots:binomial pmf

• Interarrival time pmf: geometric pmf

• Time to k arrivals: Pascal pmf

• Memorylessness

Definition of the Poisson process

x x x xx x xx x xx

Time0

t1

t2

t3 !

• P (k, τ) = Prob. of k arrivals in intervalof duration τ

• Assumptions:

– Numbers of arrivals in disjoint time in-tervals are independent

– For VERY small δ:

P (k, δ) ≈

⎧⎪⎨

⎪⎩

1 − λδ if k = 0λδ if k = 10 if k > 1

– λ = “arrival rate”

PMF of Number of Arrivals N

P (k, τ) =(λτ)ke−λτ

k!, k = 0,1, . . .

• E[N ] = λτ

• σ2N = λτ

• MN(s) = eλt(es−1)

Example: You get email according to aPoisson process at a rate of λ = 0.4 mes-sages per hour. You check your email everythirty minutes.

– Prob(no new messages)=

– Prob(one new message)=

• Finely discretize [0, t]: approximately Bernoulli

• Nt (of discrete approximation): binomial


Simeon Denis Poisson

• Deaths from horse kicks in the Priussian army (1898)





LECTURE 16

The Poisson process


Lecture outline






Bernoulli review





• Memorylessness


x x x xx x xx x xx

Time0

t1

t2

t3 !


• Assumptions:



P (k, δ) ≈

⎧⎪⎨

⎪⎩

1 − λδ if k = 0λδ if k = 10 if k > 1




k!, k = 0,1, . . .

• E[N ] = λτ

• σ2N = λτ








Simeon Denis Poisson (1781-1840)





LECTURE 16

The Poisson process


Lecture outline






Bernoulli review





• Memorylessness


x x x xx x xx x xx

Time0

t1

t2

t3 !


• Assumptions:



P (k, δ) ≈

⎧⎪⎨

⎪⎩

1 − λδ if k = 0λδ if k = 10 if k > 1




k!, k = 0,1, . . .

• E[N ] = λτ

• σ2N = λτ








• Deaths from horse kicks in the Prussian army (1898)





LECTURE 16

The Poisson process


Lecture outline






Bernoulli review





• Memorylessness


x x x xx x xx x xx

Time0

t1

t2

t3 !


• Assumptions:



P (k, δ) ≈

⎧⎪⎨

⎪⎩

1 − λδ if k = 0λδ if k = 10 if k > 1




k!, k = 0,1, . . .

• E[N ] = λτ

• σ2N = λτ













LECTURE 16

The Poisson process


Lecture outline






Bernoulli review





• Memorylessness


x x x xx x xx x xx

Time0

t1

t2

t3 !


• Assumptions:



P (k, δ) ≈

⎧⎪⎨

⎪⎩

1 − λδ if k = 0λδ if k = 10 if k > 1




k!, k = 0,1, . . .

• E[N ] = λτ

• σ2N = λτ













LECTURE 16

The Poisson process


Lecture outline






Bernoulli review





• Memorylessness


x x x xx x xx x xx

Time0

t1

t2

t3 !


• Assumptions:



P (k, δ) ≈

⎧⎪⎨

⎪⎩

1 − λδ if k = 0λδ if k = 10 if k > 1




k!, k = 0,1, . . .

• E[N ] = λτ

• σ2N = λτ













LECTURE 16

The Poisson process


Lecture outline






Bernoulli review





• Memorylessness


x x x xx x xx x xx

Time0

t1

t2

t3 !


• Assumptions:



P (k, δ) ≈

⎧⎪⎨

⎪⎩

1 − λδ if k = 0λδ if k = 10 if k > 1




k!, k = 0,1, . . .

• E[N ] = λτ

• σ2N = λτ











• Particle emissions and radioactive decay



LECTURE 16

The Poisson process


Lecture outline






Bernoulli review





• Memorylessness


x x x xx x xx x xx

Time0

t1

t2

t3 !


• Assumptions:



P (k, δ) ≈

⎧⎪⎨

⎪⎩

1 − λδ if k = 0λδ if k = 10 if k > 1




k!, k = 0,1, . . .

• E[N ] = λτ

• σ2N = λτ








Bernoulli Poisson

• Independence

• Constant p at each slot

LECTURE 16

The Poisson process


Lecture outline






Bernoulli review





• Memorylessness


x x x xx x xx x xx

Time0

t1

t2

t3 !


• Assumptions:



P (k, δ) ≈

⎧⎪⎨

⎪⎩

1 − λδ if k = 0λδ if k = 10 if k > 1




k!, k = 0,1, . . .

• E[N ] = λτ

• σ2N = λτ





• Time homogeneity:

P (k, τ) = Prob. of k arrivals in interval of duration τ

• Numbers of arrivals in disjoint timeintervals are independent


Bernoulli Poisson time

• Independence


LECTURE 16

The Poisson process


Lecture outline






Bernoulli review





• Memorylessness

of the Poisson process

x x x xx x xx x xx

Time0

t1

t2

t3 !


• Assumptions:



P (k, δ) ≈

⎧⎪⎨

⎪⎩

1 − λδ if k = 0λδ if k = 10 if k > 1




k!, k = 0,1, . . .

• E[N ] = λτ

• σ2N = λτ










• Independence


LECTURE 16

The Poisson process


Lecture outline






Bernoulli review





• Memorylessness


x x x xx x xx x xx

Time0

t1

t2

t3 !


• Assumptions:



P (k, δ) ≈

⎧⎪⎨

⎪⎩

1 − λδ if k = 0λδ if k = 10 if k > 1




k!, k = 0,1, . . .

• E[N ] = λτ

• σ2N = λτ










• Independence


LECTURE 16

The Poisson process


Lecture outline






Bernoulli review





• Memorylessness


x x x xx x xx x xx

Time0

t1

t2

t3 !


• Assumptions:



P (k, δ) ≈

⎧⎪⎨

⎪⎩

1 − λδ if k = 0λδ if k = 10 if k > 1




k!, k = 0,1, . . .

• E[N ] = λτ

• σ2N = λτ










• Independence


LECTURE 16

The Poisson process


Lecture outline






Bernoulli review





• Memorylessness


x x x xx x xx x xx

Time0

t1

t2

t3 !


• Assumptions:



P (k, δ) ≈

⎧⎪⎨

⎪⎩

1 − λδ if k = 0λδ if k = 10 if k > 1




k!, k = 0,1, . . .

• E[N ] = λτ

• σ2N = λτ









DefinitionDefinitionDefinitionDefinition time × × × ×


• Particle emissions and radioactive decay

Simeon Denis Poisson • Photon arrivals from a weak source (1781-1840)

(This image is in the public domain.• Financial market shocks Source: Wikipedia)


3

https://en.wikipedia.org/wiki/Sim%C3%A9on_Denis_Poisson#/media/File:Simeon_Poisson.jpg

M an and variance o the ber of arr vals

(k,r)- ( (AT ke - AT

-k)- ---r k! ' k - , 1, ...

NT ~ Bi omial(n,p)

n - r/d, p - A8 0(8 2 )

E[Nr] = Ar

var(N,,.) - Ar

Exampl

• You get email according to , a oisson process, at a rate of ,x = 5 messages per hour.

• Mean and variance of mai s received during a day -

• P(one ew message i the next hour) -

(Ar)ke-,\ 7

(k,r) - ---- , k!

k - , 1, ....

E[Nr = AT

var(Nr) = AT

• P(exactly two messages duri g each of the next thre ,e ours) -

e first arrival

• Find t e CDF: P(T1 < t) =

IT (t) , - At · Ae '

Expone tial(,\)

or >

Memorylessness. conditioned on T1 > t, the PD of T1 - t is again exponent·a1

( \. )k -AT P(k r) - AT e

' kl ' k - 0, 1, ..

h k

kl ' k- 1 ,1, ..

• Can derive its PDF by fi st finding th ,e CDF

• More intuitive argument:

k-2

ra

y

Memorylessness and the fresh-start property

• Analogous to the properties for the Bernoulli process

- plausible, given the relation between the two processes

- use intuitive reasoning

- can be proved rigorously

xamp e: 01sson fishi g

• ish are caught as a Po·sson p ocess, A= 0.6/ho r

t·s · fo·r two hours;

if you c.aught at least one t·sh, stop

e se continue unt·1 first r·sh is ca ugh

P(fish fo mo e t an two ho rs)=

I

I

X )(

P(fish for more than two a d ess than five hours)=

>< I 2

I )(

)

time

)

time

( '- )k -AT P(k T) - /\.T e

' k!

E[N 7 ] - AT






P(catch at least two t·s )-

I

I

X )(

E[f t re fis ing time I already fished for three ours]=

>< I 2

I

)

time

)( )

time

( '- )k -AT P(k T) - /\.T e

' k!

E[N 7 ] - AT






E(tota fishing t·me]=

E[n mber of fish =

I

I

X )( >< I 2

I )(

)

time

)

time

( '- )k -AT P(k T) - /\.T e

' k!

E[N 7 ] - AT

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Introduction to Probability: Lecture 22: The Poisson ... · The Poisson process • Readings: Start Section 5.2. Lecture outline • Review of Bernoulli process • Deﬁnition of