Let us summarize what we have discovered so far.
Tableau 10, From polls to bombs.
Part 1, deals with the polls portion of the segment and
introduces the binomial distribution.
The basic element, the building block, off our entire classes.
Was to form a model the toss of a coin.
And for this we go back to Jacob Bernoulli and
his publication of Ars Conjectandi in 1713.
The first time where a formal understanding of a model for
coining tossing became clear.
As an important and
fundamental building block in the understanding of chance processes.
So accordingly, a Bernoulli trial corresponds to a formal coin toss.
The abstract outcome, let's say x, is a number 0 or 1,
1 connoting a head or success, 0 a failure or a tail.
We allow the coin to be biased, but some say success probability p,
and the corresponding failure probability q, equal to 1 minus p.
Of course, a humble coin toss by itself is not very interesting.
We now build upon this, and through the medium of repeated independent trials.
To construct a model of a pole, or of a random sample.
So, accordingly, suppose X1, X2, X3 through Xn,
constitute repeated Bernoulli trials.
Repeated independent trials, corresponding to the tosses of a coin,
whose success probability is p.
At this point the sample space and
indeed the probability splace, space is, is transparent.
The sample space is a sequence of n, 0s and 1s.
To which we attach atomic measure obtained by multiplying probabilities of 0s and 1s.
That's the product measure.
We immediately, from this rich sample space, abstract out a key element.
A chance driven process of particular interest to us.
The accumulated successes in these n tosses is denoted by s sub n.
And of course, is now just a sum of the values X1 through Xn.
At this point, we have created a new chance process from
the underlying chance process of repeated trials.
This new process gives you a value, an idealized outcome S sub n.
Which takes values in a new sample space if you will of integers 0, 1, 2,
3, up till n.
Now naturally enough,
we are interested in what is the probability measure
that is inherited by this new chance variable, the accumulated successes.
And this is indeed.
Is exactly the binomial distribution with two parameters,
a number of trials in, and the success probability peak.
With those held fixed,
the number of successes ranges over a sample space from 0 through n.
With corresponding atomic mass probabilities,
the atomic mass function is given by exactly the binomial distribution.
And our exploration shows us that we understand
now what the distribution looks like.
It got a unimodel character.
It increases to a maximum value in the vicinity of n times p.
It has a center of mass, an expectation which is exactly n times p.
And it has a variance, and expected squared deviation from the center
which is exactly n times p times q, or
equivalently as standard deviation sigma, which is a square root of npq.
In other words, as n increases.
The expected center moves proportional to n, but the expected spread
moves only proportional to the square root of n around the center.
You get increasing concentration around the center.
So we could've done this from a purely abstract point of view, just.
Imagine this metaphorical experiment, do the algebraic calculations, but
of course, this is dry and boring.
What makes this come alive is the rich panoply of applications
of this beautiful simple underlying model.
The applications we have seen, span every large range.
From cards, to earned problems, to the throw of a die,
of course a toss of a coin, to more interesting and complex settings, such as,
medical testing of serum and vaccines and the process by which drugs are approved.
To understanding,
estimating some populations of large populations via small samples.
We are now in, in the province of statistical testing.
And in this province, we now have this range panoply of applications.
Who would have thought that the humble coin toss had this richness within it?
Only experience will convince a student that
the simple coin toss carries within it an inexhaustible source of inspiration.
And these applications are a small shop window of some of the consequences and
directions that one can go into when one explores coin tossing.
For our purposes now, what we've done is,
obtained a distributional form for an accumulated number of successes and
found its applicability in a wide range of applications.
Quo vadis?
Where does one go from here?
Well, we now have an explicit mathematical form, we have some understanding of it.
We have some, a feeling for it's application domain.
But if one tends to something like poles we have,
via Fisher's maximum likelihood principle, a principled procedure for
making guesses about sized of underlying sub-populations.
But what we don't yet
have is any guarantee about the quality of this procedure.
Yes we now have some reason to believe that this is a good way to do it, but
how good is it really?
How expensive is it, and what are the chances of error?
Of course these questions are still waiting and looming for us.
We shall return to these questions and these will be our concluding lectures
in this course, and that will take us into the marvelous,
remarkable world, the fabulous limit laws, but
before we do that I want to take one small digression.
And this connects us to a curious, mysterious
identity in approximation discovered by Siméon Denis Poisson in 1837.
And that discovery, connected to the binomial,
a rich new direction, leading to the province of rare events,
arrival processes, and knitting it all together.
The second of the fundamental distributions in chance,
the Pearson distribution.
So, we shall come back in Tableau ten, Part two.
With the Poisson distribution, and
we'll see its applications in a plethora of beguiling areas,
including bombs and cues.
Once we've done that, we shall come back to the binomial.
And return to the question of how good our pulling mechanism really is?