Download Probability on Finite Sets - Engr207b: Linear Control Systems 2

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Probability on Finite Sets
2 - Probability on Finite Sets
• Modeling physical phenomena
• Probability on finite sets
• The sample space and the pmf
• Events
• Unions, intersections and complements
• The axioms of probability
• Partitions
• Conditional probability
• Independence
• Dependent events
S. Lall, Stanford
2011.01.04.01
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Probability on Finite Sets
S. Lall, Stanford
2011.01.04.01
Modeling Physical Phenomena
• The deterministic way: write differential equations
Differential equations are constructed from
• physical principles
• experiments
Use the model to predict the approximate behavior of the system
• The probabilistic way: specify the probability of events
Probabilities are derived from
• physical principles, e.g., time symmetry for noise
• experiments; i.e., observed frequency of events
Again, we use the model to predict the approximate behavior of the system
e.g., in approximately 95% of trials the hovercraft will deviate less than 3m from the
given trajectory
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Probability on Finite Sets
S. Lall, Stanford
2011.01.04.01
Probability on Finite Sets
• The sample space is a finite set Ω; it’s elements are called outcomes. Exactly one
outcome occurs in every experiment.
• Function p : Ω → [0, 1] is called a probability mass function (pmf) if
X
p(a) ≥ 0 for all a ∈ Ω
and
p(a) = 1
a∈Ω
Then p(a) is the probability that outcome a ∈ Ω occurs
sum of three dice
product of two dice
0.14
0.12
0.12
0.1
0.1
0.08
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p(a) 0.06
p(a)
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Probability on Finite Sets
S. Lall, Stanford
2011.01.04.01
Events
An event is a subset of Ω
For example, if Ω = {1, . . . , 2n}, the following are events
• A = { 2, 4, 6, . . . , 2n }, which we would call the event that the outcome is even
• A = { x ∈ Ω | x ≥ 32 }, which we would call the event that the outcome is ≥ 32
The probability of an event is
X
Prob(A) =
p(b)
b∈A
Prob : 2Ω → [0, 1] is called a probability measure
Example: Prob(A) = 0.275, Prob(B) = 0.65
A
B
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Probability on Finite Sets
S. Lall, Stanford
Unions, Intersections and Complements
For any sets A, B ⊂ Ω we have
Prob(A ∪ B) = Prob(A) + Prob(B) − Prob(A ∩ B)
We interpret
A∪B
A∩B
Ac = { b ∈ Ω | b 6∈ A }
is the event that A or B happens
is the event that A and B happens
is the event that A does not happen
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Probability on Finite Sets
S. Lall, Stanford
Notation
Notice that Prob really depends on
• the sample space Ω
• and the probability mass function p
Sometimes we will write
Prob(A)
Ω, p
to specify which Ω and p are being used
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Probability on Finite Sets
S. Lall, Stanford
2011.01.04.01
Axioms of Probability
We have for all A, B ⊂ Ω
(i) Prob(A) ≥ 0
(ii) Prob(Ω) = 1
(iii) if A ∩ B = ∅ then Prob(A ∪ B) = Prob(A) + Prob(B)
• The above three conditions are called the axioms of probability for finite sets Ω
• If Prob : 2Ω → R satisfies the above, then we can construct a probability mass
function via
¡ ¢
p(b) = Prob {b}
for all b ∈ Ω
and p will be positive and sum to one as required.
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Probability on Finite Sets
S. Lall, Stanford
2011.01.04.01
Partitions
The set of events A1, A2, . . . , An is called a partition of Ω if
Ai ∩ Aj = ∅
for all i 6= j
A1 ∪ A2 ∪ · · · ∪ An = Ω
called mutually exclusive
called collectively exhaustive
Then for any B ⊂ Ω we have
Prob(B) =
n
X
Prob(B ∩ Ai)
i=1
called the Law of Total Probability
A1
A2
A4
A3
B
A5
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Probability on Finite Sets
S. Lall, Stanford
2011.01.04.01
Conditional Probability
Suppose A and B are events, and Prob(B) 6= 0. Define the conditional probability of A
given B by
Prob(A ∩ B)
Prob(A | B) =
Prob(B)
Example: suppose B = { x ∈ Ω | x ≥ 10}
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If we perform many repeated experiments, and throw away all x 6∈ B, then the observed
frequency of outcomes x ∈ B will increase.
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Probability on Finite Sets
S. Lall, Stanford
Conditional Probability
Conditioning defines a new probability mass function on Ω.
The conditional pmf is



p(a)
if a ∈ B
p2(a) = Prob(B)

0
otherwise
Then we have, for any A ⊂ Ω
Prob(A | B) = Prob(A)
Ω, p
Ω, p2
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Probability on Finite Sets
S. Lall, Stanford
2011.01.04.01
Independence
Two events A and B are called independent if
Prob(A ∩ B) = Prob(A) Prob(B)
• If Prob(B) 6= 0 this is equivalent to
Prob(A | B) = Prob(A)
• if A and B are dependent, then knowing whether event A occurs also gives information regarding event B
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Probability on Finite Sets
S. Lall, Stanford
2011.01.04.01
Independence
Events A and B are independent if and only if
rank(M ) = 1 where
·
¸
c
Prob(A ∩ B)
Prob(A ∩ B )
M=
Prob(Ac ∩ B) Prob(Ac ∩ B c)
A
B
M is called the joint probability matrix.
• A and B are independent means the probabilities of A occurring do not change when
we discard those outcomes when B occurs.
• The probabilities of A and Ac occurring are the row sums
·
¸
Prob(A)
= M1
Prob(Ac)
When rank(M ) = 1, each column is some multiple of M 1
·
¸
¤
Prob(A) £
c
Prob(B) Prob(B )
M=
Prob(Ac)
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Probability on Finite Sets
S. Lall, Stanford
2011.01.04.01
Example: two dice
Two dice. Pick sample space
©
ª
Ω = (ω1, ω2) | ωi ∈ {1, 2, . . . , 6}
6
5
4
A
3
Two events are
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• the sum is greater than 5
©
ª
A = ω ∈ Ω | ω1 + ω2 > 5
1
• the first dice is greater than 3
©
ª
B = ω ∈ Ω | ω1 > 3
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Probability on Finite Sets
S. Lall, Stanford
2011.01.04.01
Example: two dice
By measuring B, we have information about A, because
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Prob(A) =
36
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Prob(A | B) =
18
6
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A
3
2
1
• This is an example of estimation
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• By measuring one random quantity, we have information about another
• More refined questions: what is the conditional distribution of the sum? What should we pick as an estimate?
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• Later we will see problems of the form
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y = Ax + w
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B
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2
w is random, we measure y, and would like to know x
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