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Probability Theory and Measure:
Lecture III
I.
Uniform Probability Measure:
A.
I think that Bieren’s discussion of the uniform probability measure provides a
firm basis for the concept of probability measure.
1.
First, we follow the conceptual discussion of placing ten balls numbered 0
through 9 into a container. Next, we draw out an infinite sequence of balls
out of the container, replacing the ball each time.
2.
In
Excel,
we
can
mimic
this
sequence
using
the
function
floor(rand()*10,1). This process will give a sequence of random numbers
such as:
Table 1. Random Draws of Single Digits
Ball
Drawn
Draw 1
Draw 2
Draw 3
1
7
0
3
2
4
2
0
3
1
9
2
4
4
6
2
5
8
4
0
6
3
5
4
Taking each column, we can generate three random numbers {0.741483,
0.029645, 0.302204}. Note that each of these sequences are contained in
the unit interval
0,1
. The primary point of the demonstration is that
the number draw (
0,1
x
) is a probability measure.
a.
Taking
0.741483
x
as
the
example,
we
want
to
prove
that
0,
0.741483
0.741483
Px
. To do this we want to work out
the probability of drawing a number less than 0.741483.
b.
As a starting point, what is the probability of drawing the first
number in Table 1 less than 7, it is 7 ~{0,1,2,3,4,5,6}. Thus, without
consider the second number, the probability of drawing a number
less than 0.741483 is somewhat greater than 7/10.
c.
Next, we consider drawing a second number given that the first
number drawn is greater than or equal to 7. Now, we are interested
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View Full DocumentAEB 6933 Mathematical Statistics for Food and Resource Economics
Lecture III
Professor Charles B. Moss
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 Fall '09
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