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    11.1.1 Counting Processes: 0 (N (T), T (0, ) ) 0 0 S T N (T) N(S) (S, T)

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    Mirko Mazzon
    Counting processes model the number of occurrences of events over time. A counting process N(t) represents the number of events by time t, starting from 0 at time 0. Counting processes have three key properties: 1) N(0) = 0, 2) N(t) is a non-negative integer for all t, and 3) N(t) - N(s) counts events between times s and t. Counting processes are used to model arrivals like customers or accidents. A counting process has independent increments if the number of arrivals in non-overlapping time intervals are independent. It has stationary increments if the distribution of arrivals in an interval depends only on the interval length, not its position

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    11.1.1 Counting Processes: 0 (N (T), T (0, ) ) 0 0 S T N (T) N(S) (S, T)

    Uploaded by

    Mirko Mazzon
    67 views2 pages
    Counting processes model the number of occurrences of events over time. A counting process N(t) represents the number of events by time t, starting from 0 at time 0. Counting processes have three key properties: 1) N(0) = 0, 2) N(t) is a non-negative integer for all t, and 3) N(t) - N(s) counts events between times s and t. Counting processes are used to model arrivals like customers or accidents. A counting process has independent increments if the number of arrivals in non-overlapping time intervals are independent. It has stationary increments if the distribution of arrivals in an interval depends only on the interval length, not its position

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    Counting Processes

    Original Title

    Counting Processes

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    Counting processes model the number of occurrences of events over time. A counting process N(t) represents the number of events by time t, starting from 0 at time 0. Counting processes have three key properties: 1) N(0) = 0, 2) N(t) is a non-negative integer for all t, and 3) N(t) - N(s) counts events between times s and t. Counting processes are used to model arrivals like customers or accidents. A counting process has independent increments if the number of arrivals in non-overlapping time intervals are independent. It has stationary increments if the distribution of arrivals in an interval depends only on the interval length, not its position

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    67 views2 pages

    11.1.1 Counting Processes: 0 (N (T), T (0, ) ) 0 0 S T N (T) N(S) (S, T)

    Uploaded by

    Mirko Mazzon
    Counting processes model the number of occurrences of events over time. A counting process N(t) represents the number of events by time t, starting from 0 at time 0. Counting processes have three key properties: 1) N(0) = 0, 2) N(t) is a non-negative integer for all t, and 3) N(t) - N(s) counts events between times s and t. Counting processes are used to model arrivals like customers or accidents. A counting process has independent increments if the number of arrivals in non-overlapping time intervals are independent. It has stationary increments if the distribution of arrivals in an interval depends only on the interval length, not its position

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    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 2

    Counting Processes

    17/07/16, 17:10

    11.1.1 Counting Processes


    In some problems, we count the occurrences of some types of events. In such scenarios, we are dealing with a
    counting process. For example, you might have a random process N(t) that shows the number of customers
    who arrive at a supermarket by time t starting from time 0. For such a processes, we usually assume
    N(0) = 0 , so as time passes and customers arrive, N(t) takes positive integer values.
    Definition 11.1
    A random process {N(t), t [0, )} is said to be a counting process if N(t) is the number of events
    occurred from time 0 up to and including time t . For a counting process, we assume
    1. N(0) = 0 ;
    2. N(t) {0, 1, 2, }, for all t [0, ) ;
    3. for 0 s < t, N(t) N(s) shows the number of events that occur in the interval (s, t] .

    Since counting processes have been used to model arrivals (such as the supermarket example above), we
    usually refer to the occurrence of each event as an "arrival". For example, if N(t) is the number of accidents in
    a city up to time t , we still refer to each accident as an arrival. Figure 11.1 shows a possible realization and the
    corresponding sample function of a counting process.

    Figure 11.1 - A possible realization and the corresponding sample path of a counting process..
    By the above definition, the only sources of randomness are the arrival times Ti . Before introducing the
    Poisson process, we would like to provide two definitions.
    Definition 11.2
    Let {X(t), t [0, )} be a continuous-time random process. We say that X(t) has independent increments
    if, for all 0 t1 < t2 < t3 < tn , the random variables
    https://www.probabilitycourse.com/chapter11/11_1_1_counting_processes.php

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    Counting Processes

    17/07/16, 17:10

    X(t2 ) X(t1 ), X(t3 ) X(t2 ), , X(tn ) X(tn1 )


    are independent.
    Note that for a counting process, N(ti ) N(ti1 ) is the number of arrivals in the interval (ti1 , ti ]. Thus, a
    counting process has independent increments if the numbers of arrivals in non-overlapping (disjoint) intervals

    (t1 , t2 ], (t2 , t3 ], , (tn1 , tn ]


    are independent. Having independent increments simplifies analysis of a counting process. For example,
    suppose that we would like to find the probability of having 2 arrivals in the interval (1, 2] , and 3 arrivals in
    the interval (3, 5] . Since the two intervals (1, 2] and (3, 5] are disjoint, we can write

    P 2 arrivals in (1, 2] and 3 arrivals in (3, 5] =


    (
    )

    P 2 arrivals in (1, 2] P 3 arrivals in (3, 5] .


    (
    ) (
    )

    Here is another useful definition.


    Definition 11.3
    Let {X(t), t [0, )} be a continuous-time random process. We say that X(t) has stationary increments if,
    for all t2 > t1 0 , and all r > 0 , the two random variables X(t2 ) X(t1 ) and X(t2 + r) X(t1 + r)
    have the same distributions. In other words, the distribution of the difference depends only on the length of the
    interval (t1 , t2 ] , and not on the exact location of the interval on the real line.
    Note that for a counting process N(t) , N(t2 ) N(t1 ) is the number of arrivals in the interval (t1 , t2 ] . We
    also assume N(0) = 0 . Therefore, a counting process has stationary increments if for all t2 > t1 0 ,
    N(t2 ) N(t1 ) has the same distribution as N(t2 t1 ). This means that the distribution of the number of
    arrivals in any interval depends only on the length of the interval, and not on the exact location of the interval
    on the real line.
    A counting process has independent increments if the numbers of arrivals in non-overlapping (disjoint)
    intervals are independent.
    A counting process has stationary increments if, for all t2
    distribution as N(t2 t1 ) .
    previous
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    https://www.probabilitycourse.com/chapter11/11_1_1_counting_processes.php

    > t1 0 , N(t2 ) N(t1 ) has the same

    Pagina 2 di 2

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