Conditional Probability and Distribution
Functions: Lecture IV
Charles B. Moss
August 27, 2010
I. Conditional Probability and Independence
A. In order to defne the concept oF a conditional probability it is
necessary to defne joint and marginal probabilities.
1. The joint probability is the probability oF a particular combi
nation oF two or more random variables.
2. Taking the role oF two die as an example, the probability oF
rolling a 4 on one die and a 6 on the other die is 1/36.
3. There are 36 possible outcomes oF the two die
{
1
,
1
}
,
{
1
,
2
}
,
··· {
2
,
1
}
,
{
2
,
2
}
,
6
,
6
}
.
4. ThereFore the probability oF a
{
4
,
6
}
given that the die are
Fair is 1
/
36.
B. The marginal probability is the probability one oF the random
variables irrespective oF the outcome oF the other variable.
1. Going back to the die example, there are six di±erent rolls oF
the die where the value oF the frst die is 4
{
4
,
1
}
,
{
4
,
2
}
,
{
4
,
3
}
,
{
4
,
4
}
,
{
4
,
5
}
,
{
4
,
6
}
(1)
2. Hence, again assume that the die are Fair the marginal prob
ability oF
x
1
=4is
1
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View Full DocumentAEB 6182 Agricultural Risk Analysis and Decision Making
Professor Charles B. Moss
Lecture IV
Fall 2010
P
[
x
1
=4
]=
P
[
{
4
,
1
}
]+
P
[
{
4
,
2
}
P
[
{
4
,
3
}
P
[
{
4
,
4
}
P
[
{
4
,
5
}
P
[
{
4
,
6
}
]
=
1
36
+
1
36
+
1
36
+
1
36
+
1
36
+
1
36
=
6
36
=
1
6
(2)
C. The conditional probability is then the probability of one event,
such as the probability that the ±rst die is a 4, given that the
value of another random variable is known, such as the fact that
the value of the second die roll is equal to 6. In the forgoing
example, the case of the fair die, this value is 1/6.
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 Fall '08
 Weldon
 Probability theory, Professor Charles B. Moss, Agricultural Risk Analysis

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