Reivindicaciones de prioridad que implican a marcas colectivas

Imagine that we want to describe the number of phone calls arriving at some destina- tion of interest, or, perhaps, the number of automobiles passing a particular location on a busy highway. We will call the arrival of each phone call or automobile an event. Imagine that we observe the situation for some fixed peri- od, which we will represent by the letter t. (The symbol t can represent a minute, an hour, a day, or a year.) The number of calls that arrive during the time interval t is random in the sense that it is

unpredictable. Now imagine that we divide the time interval t into n equal subintervals. Each small interval of time will equal t/n units. No matter how large t is, t/n will be very short, provided that we make n large enough. To use Poisson’s distribution our random process must conform to three simple criteria:

X₁ 0

P

X

A Poisson distribution. The probabil- ity that an event will occur in the interval between 0 and x1equals the

area beneath the curve and between the p-axis and the line x = x1.

Laplace, who recognized in Poisson a great talent. After gradua- tion Laplace helped him find a teaching position at École Polytechnique. Poisson was a devoted mathematician and researcher, and he is often quoted as asserting that life is good only for two things: to study mathematics and to teach it.

Poisson wrote hundreds of scientific and mathematical papers. He made important contributions to the study of electricity, mag- netism, heat, mechanics, and several branches of mathematics including probability theory. His name was posthumously attached to a number of important discoveries, but he received accolades while he was alive as well. In fact, most of Poisson’s contributions were recognized during his life. His peers and the

 _{For a sufficiently short period—represented by the fraction}

t/n—either one event will occur or none will occur. This con-

dition rules out the possibility of two or more events’ occur- ring in a single subinterval of time. This restriction is reason- able provided we choose n so large that the time interval t/n is very small, where the meaning of large and small depends on the context of the problem.

 _{The probability of one event’s occurring in any given interval}

t/n is proportional to the length of the interval. (In other

words, if we wait twice as long we will be twice as likely to observe an event.)

 _{Whatever happens in one subinterval (for instance, whether} a phone call is received or not received) will have no influ- ence on the occurrence of an event in any other subinterval.

If these three criteria are satisfied, then the phenomenon of interest is called a Poisson process. Once it has been established that a par- ticular process is a Poisson process then mathematicians, engineers, and scientists can use all of the mathematics that has been devel- oped to describe such processes. The Poisson process has become a standard tool of the mathematician interested in probability, the net- work design engineer, and others interested in applications of proba- bility. It has even been used to predict the number of boulders of a given size per square kilometer on the Moon. Poisson processes are everywhere.

broader public knew about and were supportive of his work in science and mathematics. Poisson, however, made one important discovery of interest to us that was not widely recognized during his life. This was also his major contribution to the theory of probability. It is called the Poisson distribution.

The Poisson probability distribution was first described in

Recherches sur la probabilité des jugements en matière criminelle et en matière civile (Researches on the probability of criminal and civil

verdicts). The goal of the text is to analyze the relationship between the likelihood of conviction of the accused and the likeli- hood of the individual’s actually having committed the crime. (Estimates of this type enable one to determine approximately how many innocent people are locked away in jail. Unfortunately, they give no insight into which people are innocent.) It was during the course of his analysis that Poisson briefly described a new kind of probability curve or distribution.

Poisson’s distribution enables the user to calculate the likelihood that a certain event will occur k times in a given time interval, where k represents any whole number greater than or equal to 0. This discovery passed without much notice during Poisson’s time. Perhaps the reason it did not draw much attention was that he could not find an eye-catching application for his insight, but con- ditions have changed. Poisson processes are now widely used; Poisson distributions are, for example, employed when developing probabilistic models of telephone networks, in which they are used to predict the probability that k phone calls will arrive at a partic- ular point on the network in a given interval of time. They are also used in the design of traffic networks in a similar sort of way. (Car arrival times are studied instead of message arrival times.) Neither of these applications could have been foreseen by Poisson or his contemporaries, of course.

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