ROYAL UNIVERSITY OF PHNOM PENH Master of IT Engineering

PROBABILITY AND RANDOM PROCESSES FOR
ENGINEERING

ASSIGNMENT
Topic: BASIC RANDOM PROCESS

Group Member: 1, Chor Sophea 2, Lun Sokhemara 3, Phourn Hourheng 4, Chea Daly |

Academic year: 2014-2015

I. Introduction
Most of the time many systems are best studied using the concept of random variables where the outcome of random experiment was associated with some numerical value. And now there are many more systems are best studied using the concept of multiple random variables where the outcome of a random experiment was associated with multiple numerical values. Here we study random processes where the outcome of a random experiment is associated with a function of time [1]. Random processes are also called stochastic processes. For example, we might study the output of a digital filter being fed by some random signal. In that case, the filter output is described by observing the output waveform at random times.

Figure 1.1 The sequence of events leading to assigning a time function x(t) to the outcome of a random experiment Thus a random process assigns a random function of time as the outcome of a random experiment. Figure 1.1 graphically shows the sequence of events leading to assigning a function of time to the outcome of a random experiment. First we run the experiment, then we observe the resulting outcome. Each outcome is associated with a time function x(t). A random process X(t) is described by * The sample space S which includes all possible outcomes s of a random experiment * The sample function x(t) which is the time function associated with an outcome s. The values of the sample function could be discrete or continuous * The ensemble which is the set of all possible time functions produced by the random experiment * The time parameter t which could be continuous or discrete * The statistical dependencies among the random processes X(t) when t is changed. Based on the above descriptions, we could have four different types of random processes:
a) Discrete time, discrete value: We measure time at discrete values t = nT with n = 0, 1, 2, . . As an example, at each value of n we could observe the number of cars on the road x(n). In that case, x(n) is an integer between 0 and 10, say. Each time we perform this experiment, we would get a totally different sequence for x(n).
b) Discrete time, continuous value: We measure time at discrete values t = nT with n = 0, 1, 2, . ... As an example, at each value of n we measure the outside temperature x(n). In that case, x(n) is a real number between −30◦ and +45◦, say. Each time we perform this experiment, we would get a totally different sequence for x(n).
c) Continuous time, discrete value: We measure time as a continuous variable t. As an example, at each value of t we store an 8-bit digitized version of a recorded voice waveform x(t). In that case, x(t) is a binary number between 0 and 255, say. Each time we perform this experiment, we would get a totally different sequence for x(t).
d) Continuous time, continuous value: We measure time as a continuous variable t. As an example, at each value of t we record a voice waveform x(t). In that case, x(t) is a real number between 0 V and 5 V, say. Each time we perform this experiment, we would get a totally different sequence for x(t).

Figure. 1.2 An example of a discrete time, discrete valued random process for an observation of 10 samples where only three random functions are possible. The figure 1.2 above shows a discrete time, discrete valued random process for an observation of 10 samples where only three random functions are generated. We find that for n = 2, the values of the functions correspond to the random variable X(2). Therefore, random processes give rise to random variables when the time value t or n is fixed. This is equivalent to sampling all the random functions at the specified time value, which is equivalent to taking a vertical slice from all the functions shown in the figure. Example 1 A time function is generated by throwing a die in three consecutive throws and observing the number on the top face after each throw. Classify this random process and estimate how many sample functions are possible. This is a discrete time, discrete value process. Each sample function will be have three samples and each sample value will be from the set of integers 1 to 6. For example, one sample function might be 4, 2, 5. Using the multiplication principle for probability, the total number of possible outputs is 63 = 216.

II. Objective
In this section of the research paper we are considering about a production line for 10 kHz oscillators, the output frequency of each oscillator is a random variable W uniformly distributed between 9980 Hz and 1020 Hz. The frequencies of different oscillators are independent. The oscillator company has an order for one part in 104 oscillators with frequency between 9999 Hz and 10, 001 Hz. A technician takes one oscillator per minute from the production line and measures its exact frequency. (This test takes one minute.) The random variable Tr minutes is the elapsed time at which the technician finds r acceptable oscillators.
(a) What is p, the probability that any single oscillator has one-part-in-104 accuracy?
(b) What is E[T1] minutes, the expected time for the technician to find the first one-part-in-104 oscillator?
(c) What is the probability that the technician will find the first one-part-in-104 oscillator in exactly 20 minutes?
(d) What is E[T5], the expected time of finding the fifth one-part-in-104 oscillator?

III. Literature
A random process that is useful for modeling events occurring in time is the Poisson random process. A typical realization is shown in which the events, indicated by the "x"s, occur randomly in time. The random process, whose realization is a set of times, is called the Poisson random process. The random process that counts the number of events in the time interval [0, t], and which is denoted by N{t), is called the Poisson counting random process. It is clear that the two random processes are equivalent descriptions of the same random phenomenon. Note that N{t) is a continuous-time/discrete-valued (CTDV) random process. Also, because N(t) counts the number of events from the initial time t = 0 up to and including the time t, the value oi N{t) at a jump is iV(t+). Thus, N{t) is right-continuous (the same property as for the CDF of a discrete random variable). The motivation for the widespread use of the Poisson random process is its ability to model a wide range of physical and man-made random phenomena. Some of these are the distribution in time of radioactive counts, the arrivals of customers at a cashier, requests for service in computer networks, and calls made to a central location, to name just a few. In Chapter 5 we gave an example of the application of the Poisson PMF to the servicing of customers at a supermarket checkout. Here we examine the characteristics of a Poisson random process in more detail, paying particular attention not only to the probability of a given number of events in a time interval but also to the probability for the arrival times of those events. In order to avoid confusing the probabilistic notion of an event with the common usage, we will refer to the events shown in Figure 21.1 as arrivals.

The Poisson random process is a natural extension of a sequence of independent and identically distributed Bernoulli trials. The Poisson counting random process N{t) then becomes the extension of the binomial counting random process discussed in Example 16.5. To make this identification, consider a Bernoulli random process, which is defined as a sequence of IID Bernoulli trials, with U[n] = 1 with probability p and U[n] = 0 with probability 1 - p. Now envision a Bernoulli trial for each small time slot of width At in the interval [0, t] as shown in Figure 21.2. Thus, we will observe either a 1 with probability p or a 0 with probability 1-p for each of the M = t/At time slots. Recall that on the average we will observe Mp ones. Now if At -> 0 and M ->• oo with t = MAt held constant, we will obtain the Poisson random process as the limiting form of the Bernoulli random process. Also, recall that the number of ones in M IID Bernoulli trials is a binomial random variable. Hence, it seems reasonable that the number of arrivals in a Poisson random process should be a Poisson random variable in accordance with our results in Section 5.6. We next argue that this is indeed the case. For the binomial counting random process, thought of as one trial per time slot, we have that the number of ones in the interval [0, t] has the PMF

But as M ^ 00 and p -)• 0 with E[N{t)] = Mp being fixed, the binomial PMF becomes the Poisson PMF or N{t) ~ Pois(A'), where V = E[N(t)] = Mp. (Note that as the number of time slots M increases, we need to let p ^ 0 in order to maintain an average number of arrivals in [0,^].) Thus, replacing A' by E[N{t)], we write the Poisson PMF as

To determine E[N{t)] for use in (21.1), where t may be arbitrary, we examine Mp in the limit. Thus,

where we define A as the limit of p/At. Since A = E[N(t)]/t, we can interpret A as the average number of arrivals per second or the rate of the Poisson random process. This is a parameter that is easily specified in practice. Using this definition we have that

As mentioned previously, N{t) is the Poisson counting random process and the probability of k arrivals from t = 0 up to and including t is given by (21.2). It is a semi-infinite random process with N(0) = 0 by definition. It is possible to derive all the properties of a Poisson counting random process by employing the previous device of viewing it as the limiting form of a binomial counting random process as At —>• 0. However, it is cumbersome to do so and therefore, we present an alternative derivation that is consistent with the same basic assumptions. One advantage of viewing the Poisson random process as a limiting form is that many of its properties become more obvious by consideration of a sequence of IID Bernoulli trials. These properties are inherited from the binomial, such as, for example, the increments N{t2) — N{ti) must be independent. (Can you explain why this must be true for the binomial counting random process?)

Derivation of Poisson Counting Random Process:
We next derive the Poisson counting random process by appealing to a set of axioms that are consistent with our previous assumptions. Clearly, since the random process starts at t = 0, we assume that iV(0) = 0. Next, since the binomial counting random process has increments that are independent and stationary (Bernoulli trials are IID), we assume the same for the Poisson counting random process. Thus, for two increments we assume that the random variables / i = N{t2) — N(ti) and h = Ar(^4) — N{ts) are independent if t4 > ts > t2 > h and also have the same PDF if additionally t^ — ts = t2 — ti. Likewise, we assume this is true for all possible sets of increments. Note that ^4 > ts > ^2 > ^1 corresponds to nonoverlapping time intervals. The increments will still be independent if ^2 = h or the time intervals have a single point in common since the probability of N(t) changing at a point is zero as we will see shortly. As for the Bernoulli random process, there can be at most one arrival in each time slot. Similarly, for the Poisson counting random process we allow at most one arrival for each time slot so that

With these axioms we wish to prove that (21.2) follows. The derivation is indicative of an approach commonly used for analyzing continuous-time Markov random processes [Cox and Miller 1965] and so is of interest in its own right.

Derivation:
To begin, consider the determination of P[N{t) = 0] for an arbitrary t > 0. Then referring to Figure 21.3a we see that for no arrivals in [0, t], there must be no arrivals in [0, t — At] and also no arrivals in {t — At, t]. Therefore,

for which the solution is Po(t) = cexp(—A^), where c is an arbitrary constant. To evaluate the constant we invoke the initial condition that Po{0) = P[N(0) = 0] = 1 by Axiom 1 to yield c= 1. Thus, we have finally that
P[N(t) =0]= Po{t) = exp(-At).
Next we use the same argument to find a differential equation for Pi (t) = P[N(t) =
1] by referring to Figure 21.3b. We can either have no arrivals in [0, t — At] and one arrival in {t — At, t] or one arrival in [0, t — At] and no arrivals in (t — At, t]. These are the only possibilities since there can be at most one arrival in a time interval of length At. The two events are mutually exclusive so that

IV. Methodology
(a) Each resistor has frequency W in Hertz with uniform PDF
The probability that a test yields a one part in 104 oscillator is

(b) To find the PMF of T1, we view each oscillator test as an independent trial. A success occurs on a trial with probability p if we find a one part in 104 oscillator. The first one part in 104 oscillator is found at time T1 = t if we observe failures on trials 1,...,t−1 followed by a success on trial t. Hence,T1 has the geometric PMF

A geometric random variable with success probability p has mean 1/p. The expected time to find the first good oscillator is E[T1] = 1/p = 20 minutes.

(c) Since p = 0.05, the probability the first one part in 104 oscillator is found in exactly 20 minutes is PT1(20) = (0.95)19(0.05) = 0.0189.

(d) The time T5 required to find the 5th one part in 104 oscillator is the number of trials needed for 5 successes. T5 is a Pascal random variable. If this is not clear, see where the Pascal PMF is derived. When we are looking for 5 successes, the Pascal PMF is

Looking up the Pascal PMF in Appendix A, we find that E[T5] = 5/p = 100 minutes. The following argument is a second derivation of the mean of T5. Once we find the first one part in 104 oscillator, the number of additional trials needed to find the next one part in 104 oscillator once again has a geometric PMF with mean 1/p since each independent trial is a success with probability p. Similarly, the time required to find 5 one part in 104 oscillators is the sum of five independent geometric random variables. That is,

where each Ki is identically distributed to T1. Since the expectation of the sum equals the sum of the expectations
,

V. Result
According to the solution done in the methodology section we have found that the probability that any single oscillator has one-part-in-104 accuracy is 5% success. And on the other hand, the expected time for the technician to find the first one-part-in-104 oscillator is 20 minutes. For the other thing to find is the probability that the technician will find the first one-part-in-104 oscillator in exactly 20 minutes is 0.0189. And at the end of this result we have found that the expected time of finding the fifth one-part-in-104 oscillator is 100 minutes.

VI. Conclusion
The Poisson distribution has a strong theoretical background and very wide spectrum of practical applications. Bringing original and/or unusual cases, featuring Poisson processes, may provide opportunities for increasing students' attentiveness and interests in Statistics. One important lesson the author of this paper has learned is that presentation of statistical cases, including Poisson examples, should be accompanied and enriched by significant business, social or historical background description and discussion.

References https://onlinecourses.science.psu.edu/stat414/book/export/html/54 https://www.math.ust.hk/~maykwok/courses/ma246/04_05/04MA246EX_Ran.pdf http://web.xidian.edu.cn/cfliu/files/20121125_153153.pdf http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-262-discrete-stochastic-processes- spring-2011/course-notes/MIT6_262S11_chap02.pdf
http://www.intmath.com/counting-probability/13-poisson-probability-distribution.php