So, thus far we;ve seen that the binomial distribution arises serendipitously,
for example, in the analysis of poles,
in the testing of Sarin vaccines, in sundry problems, in the throwing of dice,
in cards, and of course in the tossing of a coin.
But if we want to go beyond
a qualitative understanding of where distribution arises.
Go beyond a subjective feeling of, well, this seems to fit the problem well.
And get to a position where we have a quantitative understanding
of what is going on.
If we, for example,
want to give certain guarantees on how well we estimate something.
We are going to have to dig deeper into the mathematical innards of
the distribution, of the particular function that it comprises the binomial.
So accordingly, let's jump right in and
start with an image that we have seen before.
So, on the screen you see examples of three binomial distributions.
We have fixed the number of trials at n equal to 100, and
varied the success probability from 0.3 to 0.5,
that is to say a fair coin, to 0.8.
And the mass functions are plotted on the screen for you.
Now when one looks at this particular graph for
these mass functions, we see that certain characteristics sort of jump out at us.
So in particular it seems to be clear that these atomic masses sort of increase
generally, reach a maximum, a crescendo, and then sort of decrease gently.
In other words, they have a unimodal character.
Something like an Arabian camel.
But of course is this metaphor a Bactrian camel would be bimodal.
And I guess an alien camel from somewhere else could be multimodal.
But we are dealing with an Arabian camel.
A single mode.
One maximum.
There are other features that are also seem to be evident
in this kind of picture.
For instance, there seems to be a centering tendency.
So pick a distribution.
Let's say for p equal to 0.5.
It looks clear that the masses are centered
around 50 successes, right at the midpoint.
And they seem to be distributed reasonably evenly on both sides.
Now, when the coin is fair, when p is 0.5 this tendency is very clear.
In fact, there's complete symmetry on both sides of the center.
But this kind of overall general symmetry
seems to be also persistent when p is 0.3 and p is 0.8,
in other words the masses are spread reasonably evenly around a central point.
There is a centering tendency.
Still then to the point that might come to our attention is that
the spread of likely values for the accumulated number
of successes is spread fairly tightly around the central area.
Yes, in a hundred tosses of a coin it is possible to get no
successes whatsoever, but the probability is miniscule, it's really tiny.
Most of the mass seems to be concentrated around the center.
And so there's an idea, a notion, of a spread or a dispersion around the center.
And finally, and this conclusion seems inescapable because of the graphic
demands of the figure, there appears to be a generic bell shape to the distribution.
If you connect the dots,
as it were, we find that a beautiful little bell curve emerges.
None of these are serendipitous or accidental.
And so we will examine each of these features one at a time but defer the last,
the bell curve, the most important of them all, for a subsequent tableau.
I mean in fact, it is so
important we should devote a concluding tableau just to that one issue.
So we'll focus for, for the nodes on our observed unimodality,
our centering tendency, and the nature of the spread around the center.