Stat 311 Introduction to Mathematical
Statistics
Zhengjun Zhang
Department of Statistics, University of Wisconsin, Madison, WI 53706, USA
September 1517, 2009
•
Mutually
exclusive
sets: No common element between any pair of sets.
•
Exhaustive
sets: The union of all sets is the sample space Ω.
•
A
and
˜
A
are mutually exclusive and exhaustive.
Long run relative frequency
•
Sometimes there is no reason to believe outcomes are equally likely.
•
Process: repeat an experiment over and over and count how many times an event
A
occurs. Assume we run
N
trials, then
P
(
A
)
≈
# of times
A
occurs
N
Benford’s law The first digits of numbers in legitimate records often follow a distribution
First digit
1
2
3
4
5
6
7
8
9
Proportion
.301
.176
.125
.097
.079
.076
.058
.051
.046
Example
Want to find the probability of new car defects
•
Sample a group of 237,412 new car owners
•
Find 2506 complains about their vehicles
•
A
=
{
a car has defects
}
P
(
A
)
≈
# complains
# surveys
=
2506
237412
= 0
.
0106
non equally likely, non long run relative frequency
0
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•
Need prior information
•
Ex: Two friends are both in love with a woman, but only one (maybe none) can marry
the woman.
•
Bayesian procedure?
Odds
•
Determine your own personal probability using your prior knowledge.
•
The odds in favor of an event or a proposition are the quantity
p/
(1

p
), where
p
is the
probability of the event or proposition. The odds against the same event are (1

p
)
/p
.
•
In gambling, the odds on display do not represent the true chances that the event will
occur, but are the amounts that the bookmaker will pay out on winning bets.
•
For example, suppose that you are willing to make a 1 dollar bet giving 2 to 1 odds that
Badgers will win. Then you are willing to pay 2 dollars if Badgers loses in return for
receiving 1 dollar if Badgers wins. This means that you think the appropriate probability
for Badgers winning is 2/3.
Example John and Mary are taking a mathematics course. The course has only three grades:
A, B, and C. The probability that John gets a B is .3. The probability that Mary gets a B
is .4.
The probability that neither gets an A but at least one gets a B is .1.
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 Fall '08
 Staff
 Statistics, Mutually Exclusive, Probability distribution, Probability theory, probability density function, density function, Cumulative distribution function

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