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Review
Properties of P
Equal probabilities
How to count.
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ENGRD 2700
Basic Engineering Probability
and Statistics
Lecture 4: Probability Models
David S. Matteson
School of Operations Research and Information Engineering
Rhodes Hall, Cornell University
Ithaca NY 14853 USA
[email protected]
January 29, 2009
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Properties of P
Equal probabilities
How to count.
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1.
Review
1.1.
Deﬁnition
A probability model has 3 components:
1. A
sample space
S
–abstract set containing a subset which can be put in 11
correspondence with the ’outcomes’ of experiment we wish to model.
2. A distinguished collection of subsets
A
of
S
which we designate as
events
.
When
S
is discrete,
A
is all subsets of
S
.
3. A
probability measure
P
. This is a rule assigning numbers between 0 and 1
to events. Formally,
P
is a function with domain
A
and range [0
,
1] such that
(a)
P
(
S
) = 1.
(b) 0
≤
P
(
A
)
≤
1
,
for all events
A
∈ A
.
(c)
(Countable) Additivity:
If
A
1
,A
2
,...
are disjoint events then
P
(
A
1
∪
A
2
∪
A
3
∪
...
) =
P
(
A
1
) +
P
(
A
2
) +
P
(
A
3
) +
....
(1)
Review
Properties of P
Equal probabilities
How to count.
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Remember:
•
A probabilist builds a probability model.
•
A statistician confronts a range of models with the intent to pick the one
which best ﬁts the data.
How many hats will you wear?
1.2.
Methods of constructing probability models.
Two common ways:
•
S
is
discrete:
If
S
=
{
s
1
,s
2
,...
}
, we suppose we have numbers
p
k
satisfying
p
k
≥
0;
∞
X
i
=1
p
k
= 1
.
Schematically
S
s
1
s
2
...
P
p
1
p
2
...
We deﬁne
P
by
P
(
A
) =
X
i
:
s
i
∈
A
P
(
{
s
i
}
) =
X
i
:
s
i
∈
A
p
i
.
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•
When
S
is
continuous, say a subset of
R
:
We suppose there is a density
function
f
(
x
) satisfying
f
(
x
)
≥
0
,
Z
∞
∞
f
(
u
)
du
= 1
.
We deﬁne
P
by
P
((
a,b
]) =
Z
b
a
f
(
u
)
du.
Where does
{
p
k
}
or
f
(
x
)
come from?
No magic: Most common reasons for specifying probabilities:
•
Symmetry. Feeling that we want to model a situation where all outcomes are
equally likely.
1.
Example.
Toss a red die and then a blue die. Then
S
=
{
(
i,j
) : 1
≤
i
≤
6; 1
≤
j
≤
6
}
has 36 outcomes. Because we feel each outcome should be equally likely,
we specify the
p
k
’s as 1/36 for each:
S
(1
,
1)
(1
,
2)
...
(6
,
6)
P
1/36
1/36
...
1/36
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2.
Example.
Pick a number at random from [0
,
1]. Then the set of out
comes is
S
= [0
,
1]
.
Symmetry implies
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This note was uploaded on 03/05/2009 for the course ENGRD 2700 taught by Professor Staff during the Spring '05 term at Cornell University (Engineering School).
 Spring '05
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