September 10, 2007
STA 282
INSTRUCTOR: WILLIAM M BRIGGS
1.
Week
1
1.1.
Logic.
We start with logic, and thinking straight. Here is a classical logical
argument, slightly reworked:
All statistics classes are boring.
STA 282 is a statistics class.
Therefore, STA 282 is boring.
The structure of this argument can be broken down: the two statements before
the horizontal line are called premises, the one after is called the conclusion. An
other way to say it is that the premises are our evidence for the conclusion. In any
case, we want to know: what is the probability that the conclusion is true? Given
the evidence listed, it is 1 (probability is a number between, and including, 0 and
1), the conclusion is certainly true. You are no doubt tempted to say that it is not
1, that is, that the conclusion is
not
certain, because statistics is nothing if not fun.
But that would be missing the point. You are not free to change out the evidence
given and insert, say, “Statistics is nothing if not fun.” You
must
assess the proba
bility given
only
the evidence presented. And the conlcusion here is
entailed
by the
premises given. Here is another argument, due (in form, at least) to David Hume
All the reality TV shows I have ob
served before have been ridiculous.
This is a (new) reality show before
me.
Therefore, this reality show will be
ridiculous.
The conclusion here does not follow from the premises; that is, the conclusion
is not certainly true. You may be surprised to learn this, but the universe is not
set up to guarantee that all reality TV shows will be ridiculous. It may be that,
for whatever unknown reason, that
this
show will not be ridiculous. The conclu
sion, then, is contingent on certain facts (about network executives, uncontrollable
weeping, viewers’ habits, etc.) of the universe being true, and any conclusion that
is contingent (on certain conditions about the universe holding) is never certainly
1
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INSTRUCTOR: WILLIAM M BRIGGS
true or certainly false. But how about the probability that the conclusion is true?
Pretty high, naturally, but not 1. One last argument
I will roll a die, which has six sides,
only one of which will show.
Just 1 side of the six is labeled “6.”.
Therefore, the side that shows will
be a “6.”
The conclusion here is also not certain, as will be plainly obvious to any of us.
And given just the evidence in the premises, the probability that the conclusion is
true is 1 in 6, or about 0.17.
For us, then,
probability is a measure of the logical relation between a
list of premises (or observation statements) and some conclusion.
This is not the only interpretation of probability.
There are many more, but
only two of them have large followings. The largest is called
frequentism
, the second
largest is called
Bayesian
, which itself is divided in two, a
subjective
and the logical
or
objective Bayesian
. The differences between the two flavors of Bayesianism are
slight, and have to do with how much probabilities are a matter of human beliefs;
subjectivists believe that all probabilities exist only in and only for human minds;
objectivists can show that most, if not all, probabilities are matters of logic. But
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 Fall '08
 briggs
 Statistics, Normal Distribution, Probability theory, probability density function, WILLIAM M BRIGGS

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