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Unformatted text preview: The examples above show that the problems of modeling a realworld experiment are often
connected with the question of creating a model for a set of experiments that are not
exactly the same and do not necessarily correspond to the notion of independent repetitions
within the model. In other words, the question is not only whether the probability model is
reasonable for a single experiment, but also whether the IID repetition model is appropriate
for multiple copies of the realworld experiment.
At least we have seen, however, that if a realworld experiment can be performed many times
with a physical isolation between performances that is well modeled by the IID repetition
model, then the relative frequencies of events in the realworld experiment correspond to
relative frequencies in the idealized IID repetition model, which correspond to probabilities
in the original model. In other words, under appropriate circumstances, the probabilities
in a model become essentially observable over many repetitions.
We will see later that our emphasis on IID repetitions was done for simplicity. There are
other models for repetitions of a basic model, such as Markov models, that we study later. 46 CHAPTER 1. INTRODUCTION AND REVIEW OF PROBABILITY These will also lead to relative frequencies approaching probabilities within the repetition
model. Thus, for repeated realworld experiments that are well modeled by these repetition
models, the real world relative frequencies approximate the probabilities in the model. 1.5.3 Statistical independence of realworld experiments We have been discussing the use of relative frequencies of an event A in a repeated realworld experiment to test Pr {A} in a probability model of that experiment. This can be
done essentially successfully if the repeated trials correpond to IID trials in the idealized
experiment. However, the statement about IID trials in the idealized experiment is a statement about probabilities in the extended n trial model. Thus, just as we tested Pr {A} by
repeated realworld trials of a single experiment, we should be able to test Pr {A1 , . . . , An }
in the nrepetition model by a much larger number of realworld repetitions taking an
ntuple at a time.
To be more speciﬁc, choose two large integers, m and n, and perform the underlying realworld experiment mn times. Partition the mn trials into m runs of n trials each. For any
given ntuple A1 , . . . , An of successive events, ﬁnd the relative frequency (over m trials of
n tuples) of the ntuple event A1 , . . . , An . This can then be used essentially to test the
probability Pr {A1 , . . . , An } in the model for n IID trials. The individual event probabilities
can also be tested, so the condition for independence can be tested.
The observant reader will note that there is a tacit assumption above that successive n tuples
can be modeled as independent, so it seems that we are simply replacing a big problem with
a bigger problem. This is not quite true, since if the trials are dependent with some given
probability model for dependent trials, then this test for independence will essentially reject
the independence hypothesis for large enough n.
Choosing models for realworld experiments is primarily a sub ject for statistics, and we will
not pursue it further except for brief discussions when treating particular application areas.
The purpose here was to treat a fundamental issue in probability theory. As stated before,
probabilities are nonobservables — they exist in the theory but are not directly measurable
in realworld experiments. We have shown that probabilities essentially become observable
in the realworld via relative frequencies over repeated trials. 1.5.4 Limitations of relative frequencies Most realworld applications that are modeled by probability models have such a large
sample space that it is impractical to conduct enough trials to choose probabilities from
relative frequencies. Even a shuﬄed deck of 52 cards would require many more than 52! ≈
8 × 1067 trials for most of the outcomes to appear even once. Thus relative frequencies can
be used to test the probability of given individual events of importance, but are usually
impractical for choosing the entire model and even more impractical for choosing a model
for repeated trials.
Since relative frequencies give us a concrete interpretation of what probability means, however, we can now rely on other approaches, such as symmetry, for modeling. From symmetry, 1.5. RELATION OF PROBABILITY MODELS TO THE REAL WORLD 47 for example, it is clear that all 52! possible arrangements of a card deck are equiprobable
after shuﬄing. This leads, for example, to the ability to calculate probabilities of diﬀerent
poker hands, etc., which are such popular exercises in elementary probability.
Another valuable modeling procedure is that of constructing a probability model where the
possible outcomes are independently chosen ntuples of outcomes in a simpler model. More
genera...
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This note was uploaded on 09/27/2010 for the course EE 229 taught by Professor R.srikant during the Spring '09 term at University of Illinois, Urbana Champaign.
 Spring '09
 R.Srikant

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