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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
• Distribution of number of arrivals
• Distribution of interarrival times
• 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
Applications of the Poisson process
Simeon Denis Poisson
• 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
• Distribution of number of arrivals
• Distribution of interarrival times
• 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
Applications of the Poisson process
Simeon Denis Poisson (1781-1840)
• 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
• Distribution of number of arrivals
• Distribution of interarrival times
• 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 (1781-1840)
• Deaths from horse kicks in the Prussian 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
• Distribution of number of arrivals
• Distribution of interarrival times
• 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
Applications of the Poisson process
Simeon Denis Poisson (1781-1840)
• Deaths from horse kicks in the Prussian army (1898)
• 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
• Distribution of number of arrivals
• Distribution of interarrival times
• 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
Applications of the Poisson process
Simeon Denis Poisson (1781-1840)
• Deaths from horse kicks in the Prussian army (1898)
• Placement of phone calls, service requests, etc.
• Particle emissions
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
• Distribution of number of arrivals
• Distribution of interarrival times
• 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
Applications of the Poisson process
Simeon Denis Poisson (1781-1840)
• Deaths from horse kicks in the Prussian army (1898)
• Placement of phone calls, service requests, etc.
• 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
• Distribution of number of arrivals
• Distribution of interarrival times
• 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
Applications of the Poisson process
Simeon Denis Poisson (1781-1840)
• Deaths from horse kicks in the Prussian army (1898)
• Placement of phone calls, service requests, etc.
• Particle emissions and radioactive decay
• 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
• Distribution of number of arrivals
• Distribution of interarrival times
• 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
Definition of the Poisson process
Bernoulli Poisson
• Independence
• Constant p at each slot
LECTURE 16
The Poisson process
• Readings: Start Section 5.2.
Lecture outline
• Review of Bernoulli process
• Definition of Poisson process
• Distribution of number of arrivals
• Distribution of interarrival times
• 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)=
• Time homogeneity:
P (k, τ) = Prob. of k arrivals in interval of duration τ
• Numbers of arrivals in disjoint timeintervals are independent
Definition of the Poisson process
Bernoulli Poisson time
• Independence
• Constant p at each slot
LECTURE 16
The Poisson process
• Readings: Start Section 5.2.
Lecture outline
• Review of Bernoulli process
• Definition of Poisson process
• Distribution of number of arrivals
• Distribution of interarrival times
• 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
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)=
• Time homogeneity:
P (k, τ) = Prob. of k arrivals in interval of duration τ
• Numbers of arrivals in disjoint timeintervals are independent
Definition of the Poisson process
Bernoulli Poisson time
• Independence
• Constant p at each slot
LECTURE 16
The Poisson process
• Readings: Start Section 5.2.
Lecture outline
• Review of Bernoulli process
• Definition of Poisson process
• Distribution of number of arrivals
• Distribution of interarrival times
• 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
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)=
• Time homogeneity:
P (k, τ) = Prob. of k arrivals in interval of duration τ
• Numbers of arrivals in disjoint timeintervals are independent
Definition of the Poisson process
Bernoulli Poisson time
• Independence
• Constant p at each slot
LECTURE 16
The Poisson process
• Readings: Start Section 5.2.
Lecture outline
• Review of Bernoulli process
• Definition of Poisson process
• Distribution of number of arrivals
• Distribution of interarrival times
• 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
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)=
• Time homogeneity:
P (k, τ) = Prob. of k arrivals in interval of duration τ
• Numbers of arrivals in disjoint timeintervals are independent
Definition of the Poisson process
Bernoulli Poisson time
• Independence
• Constant p at each slot
LECTURE 16
The Poisson process
• Readings: Start Section 5.2.
Lecture outline
• Review of Bernoulli process
• Definition of Poisson process
• Distribution of number of arrivals
• Distribution of interarrival times
• 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
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)=
• Time homogeneity:
P (k, τ) = Prob. of k arrivals in interval of duration τ
• Numbers of arrivals in disjoint timeintervals are independent
Applications of the Poisson process
DefinitionDefinitionDefinitionDefinition time × × × ×
• Deaths from horse kicks in the Prussian army (1898)
• 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)
• Placement of phone calls, service requests, etc.
3
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
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(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
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
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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Resource: Introduction to Probability John Tsitsiklis and Patrick Jaillet
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