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Lecture 8 Basic probability laws of discrete random variables Plan of the lecture: 1. Bernoulli trial. The binomial law. 1.1 Bernoulli trial 1.2 The Bernoulli Random Variable 1.3 The Binomial Random Variable 1.4 The multinomial distribution 2. The Poisson Random Variable 3. Simplest flow of events (simple arrival) 3.1 Flow of events 3.2 Arrival process 3.3 The Poisson process Lecturer: A.S. Eremenko 2 1 Bernoulli trial. The binomial law. 1.1 Bernoulli trial In the theory of probability and statistics, a Bernoulli trial is an experiment whose outcome is random and can be either of two possible outcomes, "success" and "failure". In practice it refers to a single experiment which can have one of two possible outcomes. These events can be phrased into "yes or no" questions: Did the coin land heads? Was the newborn child a girl? Were a person's eyes green? Did a mosquito die after the area was sprayed with insecticide? Did a potential customer decide to buy a product? Did a citizen vote for a specific candidate? Did an employee vote pro-union? Therefore success and failure are labels for outcomes, and should not be construed literally. Examples of Bernoulli trials include: 1. Flipping a coin. In this context, obverse ("heads") conventionally denotes success and reverse ("tails") denotes failure. A fair coin has the probability of success by definition. 2. Rolling a die, where a six is "success" and everything else a "failure". 3. In conducting a political opinion poll, choosing a voter at random to ascertain whether that voter will vote "yes" in an upcoming referendum. 1.2 The Bernoulli Random Variable Consider the toss of a biased coin, which comes up a head with probability , and a tail with probability . The Bernoulli random variable takes the two values and , depending on whether the outcome is a head or a tail: Its PMF is 3 For all its simplicity, the Bernoulli random variable is very important. In practice, it is used to model generic probabilistic situations with just two outcomes. Furthermore, by combining multiple Bernoulli random variables, one can construct more complicated random variables. Its mean, second moment, and variance are given by the following calculations: , , . 1.3 The Binomial Random Variable A biased coin is tossed times. At each toss, the coin comes up a head with probability , and a tail with probability , independently of prior tosses. Let be the number of heads in the -toss sequence. We refer to as a binomial random variable with parameters and . The PMF of consists of the binomial probabilities: , . (Note that here and elsewhere, we simplify notation and use , instead of , to denote the experimental values of integer-valued random variables.) The normalization property , specialized to the binomial random variable, is written as . Some special cases of the binomial PMF are sketched in Fig. 1. 4 Figure 1: The PMF of a binomial random variable. If , the PMF is symmetric around . Otherwise, the PMF is skewed towards if , and towards if . Its mean, and variance are , . Mode: or . Median: or . Skewness: . Kurtosis: . Figure 2: Binomial PMF and CDF The binomial distribution is of repeated use each time a trial involves superposing independent elementary trials. In the field of traffic, the Engset problem introduces it as a limiting distribution of the number of busy lines in subscriber concentrators. 5 The binomial distribution enjoys the following property, concerning the sum of several variables: THEOREM. If two discrete random variables and have binomial distributions with parameters respectively and , the variable has a binomial distribution with parameters . 1.4 The multinomial distribution This is the generalization of the binomial law. Assume that types can be distinguished in the population, so that the population has a proportion of elements of type (with naturally ). The question is: what is the probability of observing, when extracting items, of the type , etc., of type , with . The result is: . This result has numerous applications. Imagine for instance observing a network element (e.g. a concentrator) to which various sources are connected. What is the distribution of the traffic the sources generate? Individual traffic intensity, expressed in erlangs, is distributed between and . One usually defines categories, according to the type of subscriber (professional, residential, in urban area, etc.), corresponding to traffic intensities. With two categories, the binomial distribution gives the answer. For several categories (typically: less than , between and , ..., higher than ), the distribution of the number of customers among the categories is given by the multinomial law. Finally, knowing the distribution of the different customers among the categories allows dimensioning the subscriber concentrator, using the multinomial distribution. More generally, this result holds whenever a population, composed with different sub-populations, is observed. 2 The Poisson Random Variable A Poisson random variable takes nonnegative integer values. Its PMF is given by , , 6 where is a positive parameter characterizing the PMF, see Fig. 3. It is a legitimate PMF because . To get a feel for the Poisson random variable, think of a binomial random variable with very small and very large . For example, consider the number of typos in a book with a total of words, when the probability that any one word is misspelled is very small (associate a word with a coin toss which comes a head when the word is misspelled), or the number of cars involved in accidents in a city on a given day (associate a car with a coin toss which comes a head when the car has an accident). Such a random variable can be well-modeled as a Poisson random variable. More precisely, the Poisson PMF with parameter is a good approximation for a binomial PMF with parameters and , provided , is very large, and is very small, i.e., , . In this case, using the Poisson PMF may result in simpler models and calculations. For example, let and . Then the probability of successes in trials is calculated using the binomial PMF as . Using the Poisson PMF with , this probability is approximated by 7 Figure 3: The PMF of the Poisson random variable for different values of . Note that if , then the PMF is monotonically decreasing, while if , the PMF first increases and then decreases as the value of increases. The mean of the Poisson PMF can be calculated is follows: . The last equality is obtained by noting that is the normalization property for the Poisson PMF. A similar calculation shows that the variance of a Poisson random variable is also . Mode: and if is an integer. Median: . Skewness: . Kurtosis: . Figure 4: Poisson PMF and CDF THEOREM. Let and be two Poisson variables with parameters respectively and . Then, obeys a Poisson distribution with parameter . 8 3 Simplest flow of events (simple arrival) Stemming from probability theory its application to the field of telecommunications for the solving of traffic problems has given rise to a well-known discipline: teletraffic. A stochastic process is a mathematical model of a probabilistic experiment that evolves in time and generates a sequence of numerical values. For example, a stochastic process can be used to model: (a) the sequence of daily prices of a stock; (b) the sequence of scores in a football game; (c) the sequence of failure times of a machine; (d) the sequence of hourly traffic loads at a node of a communication network; (e) the sequence of radar measurements of the position of an airplane. Each numerical value in the sequence is modeled by a random variable, so a stochastic process is simply a (finite or infinite) sequence of random variables and does not represent a major conceptual departure from our basic framework. We are still dealing with a single basic experiment that involves outcomes governed by a probability law, and random variables that inherit their probabilistic properties from that law. 3.1 Flow of events Suppose that in a time line randomly arise points – moments of appearance of some homogeneous events (for example, calls on the telephone station etc). Sequence of those moments is so called flow of events. Figure 5 Figure 6: Examples of flows of events in telecommunications 9 Figure 7: The basic service station A lot of arrival processes can be described as so called Bernoulli and Poisson processes. 3.2 Arrival process To advance in the study of traffic properties, it is necessary to look at the two components on which it depends, i.e. client arrivals and their service. Arrivals of clients at the system input are observed. To describe the phenomenon, the arrival law, the first idea is to use the time interval between successive arrivals (inter-arrival time), or the number of arrivals in a given time interval. During the time , arrivals occur. The flow intensity is then expressed as a number, the arrival rate, whose intuitive definition is: . It is then also possible to estimate the average interval between arrivals, this being the inverse of the previous quantity. Figure 7 illustrates random arrivals. Consider random arrivals in the time interval from to . The interval length is . The arrival rate is a long term average of the number of arrivals per unit time. Its mathematical symbol is and its unit is , and is given by the following equation, where is the number of arrivals in the interval of length : . Figure 7 10 Figure 8: Illustration of the terminology applied for a traffic process. Notice the difference between time intervals and instants of time. We use the terms arrival and call synonymously. The inter-arrival time, respectively the inter-departure time, are the time intervals between arrivals, respectively departures. Flow of events is called the simplest flow if it has the properties of Poisson arrival. 3.3 The Poisson process The Poisson process can be viewed as a continuous-time analog of the Bernoulli process and applies to situations where there is no natural way of dividing time into discrete periods. We consider an arrival process that evolves in continuous time, in the sense that any real number is a possible arrival time. We define , and assume that this probability is the same for all intervals of the same length . We also introduce a positive parameter to be referred to as the arrival rate or intensity of the process. Definition of the Poisson Process An arrival process is called a Poisson process with rate if it has the following properties: (a) (Time-homogeneity.) The probability of arrivals is the same for all intervals of the same length . (b) (Independence.) The number of arrivals during a particular interval is independent of the history of arrivals outside this interval. (c) (Small interval probabilities.) The probabilities satisfy , 11 . Here, and are functions of that satisfy , . The first property states that arrivals are “equally likely” at all times. The arrivals during any time interval of length are statistically the same, in the sense that they obey the same probability law. To interpret the second property, consider a particular interval , of length . The unconditional probability of arrivals during that interval is . Suppose now that we are given complete or partial information on the arrivals outside this interval. Property (b) states that this information is irrelevant: the conditional probability of arrivals during remains equal to the unconditional probability . The third property is critical. The and terms are meant to be negligible in comparison to , when the interval length is very small. They can be thought of as the terms in a Taylor series expansion of . Thus, for small , the probability of a single arrival is roughly , plus a negligible term. Similarly, for small , the probability of zero arrivals is roughly . Note that the probability of two or more arrivals is , and is negligible in comparison to as gets smaller and smaller. Figure 9: Bernoulli approximation of the Poisson process Let us now start with a fixed time interval of length and partition it into periods of length , where is a very small number; see Fig. 6. The probability of more than two arrivals 12 during any period can be neglected, because of property (c) and the preceding discussion. Different periods are independent, by property (b). Furthermore, each period has one arrival with probability approximately equal to , or zero arrivals with probability approximately equal to . Therefore, the process being studied can be approximated by a Bernoulli process, with the approximation becoming more and more accurate the smaller is chosen. Thus the probability of arrivals in time , is approximately the same as the (binomial) probability of successes in independent Bernoulli trials with success probability at each trial. While keeping the length of the interval fixed, we let the period length decrease to zero. We then note that the number of periods goes to infinity, while the product remains constant and equal to . Under these circumstances, we can see that the binomial PMF converges to a Poisson PMF with parameter . We are then led to the important conclusion that , . Note that a Taylor series expansion of , yields , , consistent with property (c). Using our earlier formulas for the mean and variance of the Poisson PMF, we obtain , , where stands for the number of arrivals during a time interval of length . These formulas are hardly surprising, since we are dealing with the limit of a binomial PMF with parameters , , mean , and variance . Let us now derive the probability law for the time of the first arrival, assuming that the process starts at time zero. Note that we have if and only if there are no arrivals during the interval . Therefore, , . 13 We then differentiate the CDF of , and obtain the PDF formula , , which shows that the time until the first arrival is exponentially distributed with parameter . We summarize this discussion in the table that follows. See also Fig. 9. Random Variables Associated with the Poisson Process and their Properties The Poisson with parameter . This is the number of arrivals in a Poisson process with rate , over an interval of length . Its PMF, mean, and variance are , , , . The exponential with parameter . This is the time until the first arrival. Its PDF, mean, and variance are , , , . Figure 10: View of the Bernoulli process as the discrete-time version of the Poisson. We discretize time in small intervals and associate each interval with a Bernoulli trial whose parameter is . The table summarizes some of the basic correspondences.