3. The Exponential Distribution and the Poisson Process.pdf
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Slide Content
3. The Exponential Distribution
and the Poisson Process
Module: Distribution Theory (4STT311)
Lecturer: Mr WJ. Dlamini
Office: SC315
and the Poisson Process
Module: Distribution Theory (4STT311)
Lecturer: Mr WJ. Dlamini
Office: SC315
INTRODUCTION
•Mathematicalmodelforareal-worldphenomenonitisalwaysnecessary
tomakecertainsimplifyingassumptions.
•Enoughsimplifyingassumptionstoenableustohandlethemathematics
butnotsomanythatthemathematicalmodelnolongerresemblesthe
real-worldphenomenon.
•Exponentialdistributionisbothrelativelyeasytoworkwithandisoften
agoodapproximationtotheactualdistribution.
•Thepropertyoftheexponentialdistributionwhichmakesiteasyto
analyseisthatitdoesnotdeterioratewithtime.
•Mathematicalmodelforareal-worldphenomenonitisalwaysnecessary
tomakecertainsimplifyingassumptions.
•Enoughsimplifyingassumptionstoenableustohandlethemathematics
butnotsomanythatthemathematicalmodelnolongerresemblesthe
real-worldphenomenon.
•Exponentialdistributionisbothrelativelyeasytoworkwithandisoften
agoodapproximationtotheactualdistribution.
•Thepropertyoftheexponentialdistributionwhichmakesiteasyto
analyseisthatitdoesnotdeterioratewithtime.
Exponential Distribution
Definition:AcontinuousrandomvariableXissaidtohavean
exponentialdistributionwithparameter??????,??????>0,ifitsprobability
densityfunctionisgivenby
�??????=ቊ
??????�
−????????????
,??????≥0
0, ??????<0
Its cumulative distribution function (c.d.f.) is given by
�??????=න
−∞
??????
����
Definition:AcontinuousrandomvariableXissaidtohavean
exponentialdistributionwithparameter??????,??????>0,ifitsprobability
densityfunctionisgivenby
�??????=ቊ
??????�
−????????????
,??????≥0
0, ??????<0
Its cumulative distribution function (c.d.f.) is given by
�??????=න
−∞
??????
����
Exponential Distribution
•The mean of the exponential distribution, �??????,is given by
�??????=න
−∞
∞
??????�??????�??????=න
0
∞
????????????�
−????????????
�??????
Integrating by part we get
�??????=−??????�
−????????????
|
0
∞
+න
0
∞
�
−????????????
�??????=
1
??????
•The moment generating function �
??????(�)of the exponential
distribution can be shown to be as follow:
�
??????�=
??????
??????−�
,for�<??????
•The mean of the exponential distribution, �??????,is given by
�??????=න
−∞
∞
??????�??????�??????=න
0
∞
????????????�
−????????????
�??????
Integrating by part we get
�??????=−??????�
−????????????
|
0
∞
+න
0
∞
�
−????????????
�??????=
1
??????
•The moment generating function �
??????(�)of the exponential
distribution can be shown to be as follow:
�
??????�=
??????
??????−�
,for�<??????
Exponential Distribution
•All moment of ??????can be obtaining through differentiating the moment
generating function of exponential distribution. For the exponential
distribution we have
�??????
2
=
2
??????
2
We may obtain the variance of ??????as:
????????????�??????=�??????
2
−�??????
2
=
1
??????
2
•All moment of ??????can be obtaining through differentiating the moment
generating function of exponential distribution. For the exponential
distribution we have
�??????
2
=
2
??????
2
We may obtain the variance of ??????as:
????????????�??????=�??????
2
−�??????
2
=
1
??????
2
Exponential Distribution
•A random variable X is said to be memory less, if
????????????>�+�??????>�=
??????[??????>�+�,??????>�]
??????[??????>�]
=????????????>�,forall�,�≥0
•A random variable X is said to be memory less, if
????????????>�+�??????>�=
??????[??????>�+�,??????>�]
??????[??????>�]
=????????????>�,forall�,�≥0
EXAMPLE 2-1
•Suppose that the amount of time one spend at the shop is
exponentially distributed with mean ten minutes.
a)What is the probability that the customer will spend more than
fifteen minutes in the shop?
b)What is the probability that the customer will spend more than
fifteen minutes in the shop given that the customer is still in the
bank after ten minutes?
•Suppose that the amount of time one spend at the shop is
exponentially distributed with mean ten minutes.
a)What is the probability that the customer will spend more than
fifteen minutes in the shop?
b)What is the probability that the customer will spend more than
fifteen minutes in the shop given that the customer is still in the
bank after ten minutes?
EXAMPLE 2-1 SOLUTION
a)Let X represents the amount of time that the customer spends in
the shop, then we have
????????????>15=�
−15??????
=�
−1.5
≈0.220.
b) Since exponential distribution is memoryless, we have that this must
be equal to the probability that an entering customer spends at least
five minutes in the shop. That is, the desired probability is
????????????>5=�
−5??????
=�
−0.5
≈0.604
a)Let X represents the amount of time that the customer spends in
the shop, then we have
????????????>15=�
−15??????
=�
−1.5
≈0.220.
b) Since exponential distribution is memoryless, we have that this must
be equal to the probability that an entering customer spends at least
five minutes in the shop. That is, the desired probability is
????????????>5=�
−5??????
=�
−0.5
≈0.604
Further Properties of the Exponential
•Let ??????
1,??????
2,??????
3,…,??????
??????be independent and identically distributed
exponential random variable with the mean Τ1??????.
•It can be shown that this has a gamma distribution with the
parameters �and ??????.
•Let us now give a verification of this result by using mathematical
induction.
•As there is nothing to prove when �=1, let us start by assuming that
??????
1+??????
2+??????
3+…+??????
??????−1has density given by
•Let ??????
1,??????
2,??????
3,…,??????
??????be independent and identically distributed
exponential random variable with the mean Τ1??????.
•It can be shown that this has a gamma distribution with the
parameters �and ??????.
•Let us now give a verification of this result by using mathematical
induction.
•As there is nothing to prove when �=1, let us start by assuming that
??????
1+??????
2+??????
3+…+??????
??????−1has density given by
Further Properties of the Exponential
�
??????1+⋯+????????????−1
�=??????�
−??????�
(??????�)
??????−2
�−2!
Hence
�
??????1+⋯+????????????
�
=න
0
∞
�
??????
??????
�−��
??????
1+⋯+??????
??????−1
���=න
0
�
??????�
−??????(�−�)
??????�
−??????�
(??????�)
??????−2
�−2!
��
Which proves the result.
�
??????1+⋯+????????????−1
�=??????�
−??????�
(??????�)
??????−2
�−2!
Hence
�
??????1+⋯+????????????
�
=න
0
∞
�
??????
??????
�−��
??????
1+⋯+??????
??????−1
���=න
0
�
??????�
−??????(�−�)
??????�
−??????�
(??????�)
??????−2
�−2!
��
Which proves the result.
Further Properties of the Exponential
•Another useful computation is to determine the probability that one
exponential random variable is smaller than another.
•Suppose that ??????
1and ??????
2are independent exponential random
variables with means Τ1??????
1and Τ1??????
2respectively; then what is
??????[??????
1<??????
2]?
•We can calculate this by conditioning on ??????
1:
????????????
1<??????
2=න
0
∞
????????????
1<??????
2??????
1=????????????
1�
−??????
1??????
�??????=
??????
1
??????
1+??????
2
•Another useful computation is to determine the probability that one
exponential random variable is smaller than another.
•Suppose that ??????
1and ??????
2are independent exponential random
variables with means Τ1??????
1and Τ1??????
2respectively; then what is
??????[??????
1<??????
2]?
•We can calculate this by conditioning on ??????
1:
????????????
1<??????
2=න
0
∞
????????????
1<??????
2??????
1=????????????
1�
−??????
1??????
�??????=
??????
1
??????
1+??????
2
EXAMPLE 2-2
•Supposeonehasastereosystemconsistingoftwomainparts,aradio
andaspeaker.Ifthelifetimeoftheradioisexponentialwithmean
10000hoursandthelifetimeofthespeakerisexponentialwithmean
5000hoursindependentoftheradio'slifetime,thenwhatisthe
probabilitythatthesystem'sfailure(whenitoccurs)willbecausedby
theradiofailing?
•Solution
Let ??????
1radio part and ??????
2speaker part. We have that ??????
1=Τ110000and
Τ??????
2=15000. The answer is as follows:
????????????
1<??????
2=
??????
1
??????
1+??????
2
=
Τ110000
Τ110000+Τ15000
=
1
3
=0.333
•Supposeonehasastereosystemconsistingoftwomainparts,aradio
andaspeaker.Ifthelifetimeoftheradioisexponentialwithmean
10000hoursandthelifetimeofthespeakerisexponentialwithmean
5000hoursindependentoftheradio'slifetime,thenwhatisthe
probabilitythatthesystem'sfailure(whenitoccurs)willbecausedby
theradiofailing?
•Solution
Let ??????
1radio part and ??????
2speaker part. We have that ??????
1=Τ110000and
Τ??????
2=15000. The answer is as follows:
????????????
1<??????
2=
??????
1
??????
1+??????
2
=
Τ110000
Τ110000+Τ15000
=
1
3
=0.333
Poisson Process
•A stochastic process {��,�≥0}is said to be a counting process if �(�)
represents the total number of “events” that have occurred up to time �.
Some example of counting processes are as follow:
1.If �(�)equals to the number cars that passes Mtunziniplaza sheshalane
by time �, then ��,�≥0is a counting process. An event of this
process will occur whenever a car passes Mtunziniplaza sheshalane.
2.If we let �(�)equals to the total number of children born by time �, then
��,�≥0is a counting process.
3.If �(�)equals the number of goals that a given Richards bay football club
has scored by time�, then ��,�≥0is a counting process. An event
of this process will occur whenever the soccer player scores a goal.
•A stochastic process {��,�≥0}is said to be a counting process if �(�)
represents the total number of “events” that have occurred up to time �.
Some example of counting processes are as follow:
1.If �(�)equals to the number cars that passes Mtunziniplaza sheshalane
by time �, then ��,�≥0is a counting process. An event of this
process will occur whenever a car passes Mtunziniplaza sheshalane.
2.If we let �(�)equals to the total number of children born by time �, then
��,�≥0is a counting process.
3.If �(�)equals the number of goals that a given Richards bay football club
has scored by time�, then ��,�≥0is a counting process. An event
of this process will occur whenever the soccer player scores a goal.
Poisson Process
•Based on this definition, we can clearly see that for a counting
process �(�)must satisfy:
i.�(�)≥0.
ii.�(�)is integer valued.
iii.If �<�,then ��≤��.
iv.For �<�,��−�(�)equals the number of events that have
occurred in the interval (�,�).
•Based on this definition, we can clearly see that for a counting
process �(�)must satisfy:
i.�(�)≥0.
ii.�(�)is integer valued.
iii.If �<�,then ��≤��.
iv.For �<�,��−�(�)equals the number of events that have
occurred in the interval (�,�).
Poisson Process
•Acountingsaidtopossessindependentincrementsifthenumbersof
eventswhichoccurindisjointtimeintervalsareindependent.
•Thenumberofeventswhichoccurredbytimesay5hours[thatis,
�(5)]mustbeindependentofthenumberofeventsoccurring
betweentimes5and10hours[thatis,�10−�(5)].
•Assumptionofindependentincrementsmightbereasonable,thatisif
webelievethatasoccerplayer’schancesofscoringagoaltodayis
independentonhowthesoccerplayerhasbeengoing.
•Acountingsaidtopossessindependentincrementsifthenumbersof
eventswhichoccurindisjointtimeintervalsareindependent.
•Thenumberofeventswhichoccurredbytimesay5hours[thatis,
�(5)]mustbeindependentofthenumberofeventsoccurring
betweentimes5and10hours[thatis,�10−�(5)].
•Assumptionofindependentincrementsmightbereasonable,thatisif
webelievethatasoccerplayer’schancesofscoringagoaltodayis
independentonhowthesoccerplayerhasbeengoing.
Poisson Process
•Acountingprocessissaidtopossessstationaryincrementsifthe
distributionofthenumberofeventswhichoccurinanytimeinterval
oftimedependsonlyonthelengthofthetimeinterval.
•Processhasstationaryincrementsifthenumberofeventsinthe
interval(�
1+�,�
2+�),thatis��
2+�−�(�
1+�)hassame
distributionasthenumberofeventsintheinterval(�
1,�
2)thatis
��
2−�(�
1)forall�
1<�
2,and�>0.
•Theassumptionofstationaryincrementswouldbereasonableif
therewerenotimesofthedayatwhichcarsweremorelikelytopass
Mtunziniplaza.
•Acountingprocessissaidtopossessstationaryincrementsifthe
distributionofthenumberofeventswhichoccurinanytimeinterval
oftimedependsonlyonthelengthofthetimeinterval.
•Processhasstationaryincrementsifthenumberofeventsinthe
interval(�
1+�,�
2+�),thatis��
2+�−�(�
1+�)hassame
distributionasthenumberofeventsintheinterval(�
1,�
2)thatis
��
2−�(�
1)forall�
1<�
2,and�>0.
•Theassumptionofstationaryincrementswouldbereasonableif
therewerenotimesofthedayatwhichcarsweremorelikelytopass
Mtunziniplaza.
Poisson Process
•Definition: The counting process {��,�≥0}is said to be a Poisson
process having rate ??????, ??????>0, if
I.��=0.
II.The process has independent increments.
III.The number of events in any time interval of length t is Poisson
distributed with mean ??????�. That is, for all �,�≥0
??????��+�−��=
�
−??????�
(??????�)
??????
�!
,�=0,1,2,…
•Definition: The counting process {��,�≥0}is said to be a Poisson
process having rate ??????, ??????>0, if
I.��=0.
II.The process has independent increments.
III.The number of events in any time interval of length t is Poisson
distributed with mean ??????�. That is, for all �,�≥0
??????��+�−��=
�
−??????�
(??????�)
??????
�!
,�=0,1,2,…
Poisson Process
•Now, from condition (iii) that a Poisson process has stationary
increments and also that
���=??????�
This explains why ??????is called the rate of the process
•One can show that if an arbitrary counting process is actually a
Poisson process.
•The conditions above need to be satisfied.
•However, it is not clear how to verify if condition (iii) is satisfied, thus
an equivalent definition of a Poisson process would be useful.
•Now, from condition (iii) that a Poisson process has stationary
increments and also that
���=??????�
This explains why ??????is called the rate of the process
•One can show that if an arbitrary counting process is actually a
Poisson process.
•The conditions above need to be satisfied.
•However, it is not clear how to verify if condition (iii) is satisfied, thus
an equivalent definition of a Poisson process would be useful.
Poisson Process
•The function �??????=??????
2
is �(ℎ)because
lim
ℎ→0
�(ℎ)
ℎ
=lim
ℎ→0
ℎ
2
ℎ
=lim
ℎ→0
ℎ=0
The function �??????=??????is not �(ℎ)since
lim
ℎ→0
�(ℎ)
ℎ
=lim
ℎ→0
ℎ
ℎ
=lim
ℎ→0
1=1≠0
•If �(.)is �(ℎ)and �(.)is �(ℎ), then so is �.+�(.). This
follows since
•lim
ℎ→0
�ℎ+�(ℎ)
ℎ
=lim
ℎ→0
�(ℎ)
ℎ
+lim
ℎ→0
�(ℎ)
ℎ
=0+0=0
•The function �??????=??????
2
is �(ℎ)because
lim
ℎ→0
�(ℎ)
ℎ
=lim
ℎ→0
ℎ
2
ℎ
=lim
ℎ→0
ℎ=0
The function �??????=??????is not �(ℎ)since
lim
ℎ→0
�(ℎ)
ℎ
=lim
ℎ→0
ℎ
ℎ
=lim
ℎ→0
1=1≠0
•If �(.)is �(ℎ)and �(.)is �(ℎ), then so is �.+�(.). This
follows since
•lim
ℎ→0
�ℎ+�(ℎ)
ℎ
=lim
ℎ→0
�(ℎ)
ℎ
+lim
ℎ→0
�(ℎ)
ℎ
=0+0=0
Poisson Process
•If �.is �(ℎ), then so is �.=��.. This follows since
lim
ℎ→0
��(ℎ)
ℎ
=�lim
ℎ→0
�(ℎ)
ℎ
=�.0=0
•Based on (iii) and (iv) we can deduce that any finite linear of
functions, each of which �(ℎ), is �(ℎ).
•We are now in a position to give an alternative definition of a Poisson
process.
•If �.is �(ℎ), then so is �.=��.. This follows since
lim
ℎ→0
��(ℎ)
ℎ
=�lim
ℎ→0
�(ℎ)
ℎ
=�.0=0
•Based on (iii) and (iv) we can deduce that any finite linear of
functions, each of which �(ℎ), is �(ℎ).
•We are now in a position to give an alternative definition of a Poisson
process.
Poisson Process
Definition: The counting process {��,�≥0}is said to be a Poisson
process having rate ??????,??????>0, if
i.�0=0.
ii.The process has stationary and independent increments.
iii.??????�ℎ=1=??????ℎ+�ℎ.
iv.??????�ℎ≥2=�ℎ.
Definition: The counting process {��,�≥0}is said to be a Poisson
process having rate ??????,??????>0, if
i.�0=0.
ii.The process has stationary and independent increments.
iii.??????�ℎ=1=??????ℎ+�ℎ.
iv.??????�ℎ≥2=�ℎ.
THEOREM 3-1
•Definition 3.3.2 and definition 3.3.4 are equivalent
Proof: We start by showing that Definition 3.3.4 implies Definition
3.3.2. In doing this, we let
??????
??????�=??????[��=�]
We derive a differential equation for ??????
0(�)in the following way:
??????
0�+ℎ=??????��+ℎ=0
•Definition 3.3.2 and definition 3.3.4 are equivalent
Proof: We start by showing that Definition 3.3.4 implies Definition
3.3.2. In doing this, we let
??????
??????�=??????[��=�]
We derive a differential equation for ??????
0(�)in the following way:
??????
0�+ℎ=??????��+ℎ=0
THEOREM 3-1
=??????[��=0,��+ℎ−��=0]
=??????��=0??????[��+ℎ−��=0]
=??????
0�1−??????ℎ+�ℎ
•The final two equations follows the assumption (ii) plus the fact that
assumption (iii) and (iv) implies that ??????�ℎ=0=1−??????ℎ+�(ℎ).
Hence
??????
0�+ℎ−??????
0(�)
ℎ
=−????????????
0�+
�(ℎ)
ℎ
Or, equivalently,
??????
0
′
(�)
??????
0(�)
=−??????
=??????[��=0,��+ℎ−��=0]
=??????��=0??????[��+ℎ−��=0]
=??????
0�1−??????ℎ+�ℎ
•The final two equations follows the assumption (ii) plus the fact that
assumption (iii) and (iv) implies that ??????�ℎ=0=1−??????ℎ+�(ℎ).
Hence
??????
0�+ℎ−??????
0(�)
ℎ
=−????????????
0�+
�(ℎ)
ℎ
Or, equivalently,
??????
0
′
(�)
??????
0(�)
=−??????
THEOREM 3-1
Which implies, by integration, that
ln??????
0�=−??????+�
Or
??????
0�=??????�
−??????�
Since ??????
00=??????�0=0=1, we arrive at, and, for �>0,
??????
??????�+ℎ=??????[��+ℎ=�]
Which implies, by integration, that
ln??????
0�=−??????+�
Or
??????
0�=??????�
−??????�
Since ??????
00=??????�0=0=1, we arrive at, and, for �>0,
??????
??????�+ℎ=??????[��+ℎ=�]
THEOREM 3-1
=??????[��=�,��+ℎ−��=0]
+??????[��=�−1,��+ℎ−��=1]
+
�=2
??????
??????[��=�−�,��+ℎ−��=�]
•However, by assumption (iv), the last tern in the preceding is �(ℎ);
hence, by using assumption (ii), we obtain
??????
??????�+ℎ=??????
??????�??????
0ℎ+??????
??????−1�??????
1ℎ+�(ℎ)
=1−??????ℎ??????
??????�+??????ℎ??????
??????−1�+�(ℎ)
=??????[��=�,��+ℎ−��=0]
+??????[��=�−1,��+ℎ−��=1]
+
�=2
??????
??????[��=�−�,��+ℎ−��=�]
•However, by assumption (iv), the last tern in the preceding is �(ℎ);
hence, by using assumption (ii), we obtain
??????
??????�+ℎ=??????
??????�??????
0ℎ+??????
??????−1�??????
1ℎ+�(ℎ)
=1−??????ℎ??????
??????�+??????ℎ??????
??????−1�+�(ℎ)
THEOREM 3-1
•Thus,
??????
??????�+ℎ−??????
??????(�)
ℎ
=−????????????
??????(�)+????????????
??????−1(�)+
�(ℎ)
ℎ
and letting ℎ→0, we get
??????
??????
′
�=−????????????
??????�+????????????
??????−1�
or equivalently,
�
??????�
??????
??????
′
�+????????????
??????�=??????�
??????�
??????
??????−1(�)
Hence
�
��
�
??????�
??????
??????�=??????�
??????�
??????
??????−1(�)
•Thus,
??????
??????�+ℎ−??????
??????(�)
ℎ
=−????????????
??????(�)+????????????
??????−1(�)+
�(ℎ)
ℎ
and letting ℎ→0, we get
??????
??????
′
�=−????????????
??????�+????????????
??????−1�
or equivalently,
�
??????�
??????
??????
′
�+????????????
??????�=??????�
??????�
??????
??????−1(�)
Hence
�
��
�
??????�
??????
??????�=??????�
??????�
??????
??????−1(�)
THEOREM 3-1
•Now, by the equation ??????
0�=�
−??????�
, we have
�
��
�
??????�
??????
1�=??????
Or
??????
1�=(??????�+�)�
−??????�
This gives the following since ??????
10=0,
To show that
??????
??????�=
�
−??????�
(??????�)
??????
�!
•Now, by the equation ??????
0�=�
−??????�
, we have
�
��
�
??????�
??????
1�=??????
Or
??????
1�=(??????�+�)�
−??????�
This gives the following since ??????
10=0,
To show that
??????
??????�=
�
−??????�
(??????�)
??????
�!
THEOREM 3-1
•We use mathematical induction and hence assume it for �−1, Then
by
�
��
�
??????�
??????
??????�=??????�
??????�
??????
??????−1�,
�
��
�
??????�
??????
??????�=
??????
??????
�
??????−1
�−1!
Or
�
??????�
??????
??????(�)=
(??????�)
??????
�!
+�
This implies the result [ as ??????
??????0=0]. This proves that Definition 3.3.4
implies Definition 3.3.2.
•We use mathematical induction and hence assume it for �−1, Then
by
�
��
�
??????�
??????
??????�=??????�
??????�
??????
??????−1�,
�
��
�
??????�
??????
??????�=
??????
??????
�
??????−1
�−1!
Or
�
??????�
??????
??????(�)=
(??????�)
??????
�!
+�
This implies the result [ as ??????
??????0=0]. This proves that Definition 3.3.4
implies Definition 3.3.2.
Interarrivaland Waiting Time Distributions
•Consider a Poisson process, and suppose we denote the time of the first
event by say �
1
•Further, for �>1, let �
??????denote the elapsed time between the (�−1)st
and the ��ℎevent.
•The sequence {�
??????,�=1,2,3,…}is called the sequence of interarrival
times.
•That is, if �
1=10and �
2=15, then the first event of the Poisson process
would have occurred at time 10 and the second event at time 15.
•We want to determine the distribution of the �
??????.
•Consider a Poisson process, and suppose we denote the time of the first
event by say �
1
•Further, for �>1, let �
??????denote the elapsed time between the (�−1)st
and the ��ℎevent.
•The sequence {�
??????,�=1,2,3,…}is called the sequence of interarrival
times.
•That is, if �
1=10and �
2=15, then the first event of the Poisson process
would have occurred at time 10 and the second event at time 15.
•We want to determine the distribution of the �
??????.
Interarrivaland Waiting Time
•We first note that the event {�
1>�}takes place if and only if no
events of the Poisson process occur in the interval [0,�]and thus,
??????�
1>�=??????��=0=�
−??????�
Hence, �
1has an exponential distribution with mean Τ1??????. Now,
??????�
2>�=�{??????�
2>��
1}
However,
•We first note that the event {�
1>�}takes place if and only if no
events of the Poisson process occur in the interval [0,�]and thus,
??????�
1>�=??????��=0=�
−??????�
Hence, �
1has an exponential distribution with mean Τ1??????. Now,
??????�
2>�=�{??????�
2>��
1}
However,
Interarrivaland Waiting Time
??????�
2>��
1=�=??????{0eventin(�,�+�]|�
1=�}
=??????{0eventsin�,�+�}
=�
−??????�
•The last two equations followed from independent and stationary
increments.
•We can conclude that also �
2is also an exponential random variable with
mean Τ1??????, and that �
2is independent of �
1.
•Replacing the same argument yields the following.
•Proposition3.3.5.1: �
??????,�=1,2,3,…are independent identically distributed
exponential random variables having mean Τ1??????
??????�
2>��
1=�=??????{0eventin(�,�+�]|�
1=�}
=??????{0eventsin�,�+�}
=�
−??????�
•The last two equations followed from independent and stationary
increments.
•We can conclude that also �
2is also an exponential random variable with
mean Τ1??????, and that �
2is independent of �
1.
•Replacing the same argument yields the following.
•Proposition3.3.5.1: �
??????,�=1,2,3,…are independent identically distributed
exponential random variables having mean Τ1??????
Interarrivaland Waiting Time
•Another quantity of interest is �
??????, the arrival time of the ��ℎevent, also
called the waiting time until the nth event. It is easily seen that
�
??????=
�=1
??????
�
�, �≥1
and hence from the proposition 3.3.5.1 and results shown in 4STT211 it
follows that �
??????has a gamma distribution with parameters �and ??????.
�
??????
??????
�=??????�
−??????�
(??????�)
??????−1
�−1!
,�≥0
•The above equation may also be derived by noting that the nth event will
occur prior to or at time �if and only if the number of event occurring by
time �is at least �. that is
•Another quantity of interest is �
??????, the arrival time of the ��ℎevent, also
called the waiting time until the nth event. It is easily seen that
�
??????=
�=1
??????
�
�, �≥1
and hence from the proposition 3.3.5.1 and results shown in 4STT211 it
follows that �
??????has a gamma distribution with parameters �and ??????.
�
??????
??????
�=??????�
−??????�
(??????�)
??????−1
�−1!
,�≥0
•The above equation may also be derived by noting that the nth event will
occur prior to or at time �if and only if the number of event occurring by
time �is at least �. that is
Interarrivaland Waiting Time
��≥�⟷�
??????≤�
And hence
�
??????
??????
�=??????�
??????≤�=??????��≥�=
�=??????
∞
�
−??????�
(??????�)
�
�!
which we can differentiate and obtain
�
??????
??????
�=−
�=??????
∞
??????�
−??????�
??????�
�
�!
+
�=??????
∞
??????�
−??????�
(??????�)
�−1
(�−1)!
��≥�⟷�
??????≤�
And hence
�
??????
??????
�=??????�
??????≤�=??????��≥�=
�=??????
∞
�
−??????�
(??????�)
�
�!
which we can differentiate and obtain
�
??????
??????
�=−
�=??????
∞
??????�
−??????�
??????�
�
�!
+
�=??????
∞
??????�
−??????�
(??????�)
�−1
(�−1)!
Interarrivaland Waiting Time
=??????�
−??????�
(??????�)
??????−1
�−1!
+
�=??????+1
∞
??????�
−??????�
(??????�)
�−1
�−1!
−
�=??????
∞
??????�
−??????�
(??????�)
�
�!
=??????�
−??????�
(??????�)
??????−1
�−1!
=??????�
−??????�
(??????�)
??????−1
�−1!
+
�=??????+1
∞
??????�
−??????�
(??????�)
�−1
�−1!
−
�=??????
∞
??????�
−??????�
(??????�)
�
�!
=??????�
−??????�
(??????�)
??????−1
�−1!
EXAMPLE 2-3
•Suppose that people immigrate into a territory at a Poisson rate ??????=
1per day.
a) What is the expected time until the tenth immigrant arrives?
b) What is the probability that the elapsed time between the tenth and
the eleventh arrival exceeds two days?
Solution
a)��
10=Τ10??????=10days
b)??????�
11>2=�
−2??????
=�
−2
≈0.1333.
•Suppose that people immigrate into a territory at a Poisson rate ??????=
1per day.
a) What is the expected time until the tenth immigrant arrives?
b) What is the probability that the elapsed time between the tenth and
the eleventh arrival exceeds two days?
Solution
a)��
10=Τ10??????=10days
b)??????�
11>2=�
−2??????
=�
−2
≈0.1333.
Interarrivaland Waiting Time
•We can define a counting process by saying that the nth event of this
process occurs at time
�
??????=�
1+�
2+�
3+⋯+�
??????
The resultant counting process [��,�≥0]
∗
will be Poisson with rate
??????.
Note: Another way of obtaining the density function of �
??????is note that
since �
??????is the time of the nth event, then it follows that
•We can define a counting process by saying that the nth event of this
process occurs at time
�
??????=�
1+�
2+�
3+⋯+�
??????
The resultant counting process [��,�≥0]
∗
will be Poisson with rate
??????.
Note: Another way of obtaining the density function of �
??????is note that
since �
??????is the time of the nth event, then it follows that
Interarrivaland Waiting Time
??????�<�
??????<�+ℎ=??????��=�−1,oneeventin�,�+ℎ+�(ℎ)
=??????��=�−1??????����??????������,�+ℎ+�(ℎ)
=�
−??????�
(??????�)
??????−1
�−1!
??????ℎ+�ℎ+�ℎ
=�
−??????�
(??????�)
??????−1
�−1!
ℎ+�ℎ
We can then divide both side by of the preceding equation by ℎand let
ℎapproach zero, we obtain
�
????????????
�=??????�
−??????�
(??????�)
??????−1
�−1!
??????�<�
??????<�+ℎ=??????��=�−1,oneeventin�,�+ℎ+�(ℎ)
=??????��=�−1??????����??????������,�+ℎ+�(ℎ)
=�
−??????�
(??????�)
??????−1
�−1!
??????ℎ+�ℎ+�ℎ
=�
−??????�
(??????�)
??????−1
�−1!
ℎ+�ℎ
We can then divide both side by of the preceding equation by ℎand let
ℎapproach zero, we obtain
�
????????????
�=??????�
−??????�
(??????�)
??????−1
�−1!
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