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ST 2133 Distribution Theory
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Cards (209)
Sample space (Ω)
The set of all possible
outcomes
of an experiment
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Event (
A
)
A set of
outcomes
which is of
interest
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Classical
probability
P(A) =
n(A) / n(Ω), where n(A) is the number
of outcomes in A and n(Ω) is the
total
number of equally likely possible
outcomes
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Event A = {sum of two dice is greater than
10
}
(
5
,
6
)
(
6
,
5
)
(
6
, 6)
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n(Ω)
Finite sample space
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The concept of
'classical
probability' was first introduced in ST104b Statistics
2
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Probability
is a
mathematical
construct
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Whenever we apply the ideas of probability to real-world situations we always make
assumptions
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Probability statements
are statements about a mathematical model, not statements about
reality
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Elementary
outcomes
Outcomes of the form (value on first
die
, value on second
die
)
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Event A = {(
5
, 6), (6,
5
), (6, 6)}
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A ⊂
Ω
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Assuming
fair dice
, all members of the sample space are
equally
likely
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n(A) =
3
, n(Ω) =
36
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P(A) =
3/36
=
1/12
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The probability calculated is a statement about a model for rolling two dice in which each of the possible outcomes is
equally
likely
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For the model to be valid, the dice must be fair such that the six outcomes from rolling each die are
equally
likely
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Slight differences in the
likelihood
of
outcomes
may be ignored for simplicity
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Tossing a fair coin repeatedly until both a
head
and a
tail
have appeared at least once
ΩH = {i tails followed by 1 head : i =
1
,
2
, ...}
ΩT = {i heads followed by
1
tail : i = 1,
2
, ...}
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P(TTH) + P(HHT) =
2
×
1/2
× 1/3 = 1/4
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Probability that a child has
blue
eyes is
0.25
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Assume
independence
between children in a family with
three
children
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P({A1, A2, A3} : at least
two
of Ai are 1 | {A1, A2, A3} : at least one of Ai is 1) =
0.2703
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P({A1, A2, 1} : at least one of A1 and
A2
are 1 | A3 = 1) =
0.4375
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σ-algebra
A collection of
subsets
of a set Ψ that satisfies: 1) ∅ ∈ B, 2) if A ∈ B then Ac ∈ B, 3) if
A1
, A2, ... ∈ B then ∪∞i=1 Ai ∈ B
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Measurable space
The pair (Ψ, B) where Ψ is a
set
and B is a
σ-algebra
defined on Ψ
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Measure
A function m: B → R+ such that: 1) m(
A
) ≥ 0 for all A ∈ B, 2) m(∅) = 0, 3) if A1, A2, ... ∈ B are
mutually
exclusive then m(∪∞i=1 Ai) = ∑∞i=1 m(Ai)
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Measure space
The triple (Ψ, B, m) where (Ψ, B) is a
measurable
space and m is a
measure
on (Ψ, B)
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Indicator function
IΩ(x) = 1 if x ∈ Ω, 0 otherwise
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Counts
and indicator functions are examples of
measures
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Probability measure
A measure P: F → [0, 1] on a
measurable
space (Ω, F) such that P(Ω) =
1
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Probability space
The triple (Ω, F, P) where (Ω, F) is a
measurable
space and P is a
probability
measure on (Ω, F)
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P(A)
is the probability that the outcome of the experiment is in
A
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P(Ac)
is the probability that the outcome of the experiment is not in
A
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P(A ∩ B) is the probability that the outcome of the experiment is in
both
A and B
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P(
A
∪ B) is the probability that the outcome of the experiment is in A or B or in
both
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Probability space
A collection of events (forming a σ-algebra on the sample space) and a
probability measure
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Probability
measures
P(
A)
= n(A)/n(
Ω
)
P(A) =
IA
(ω) for some ω ∈
Ω
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Outcomes
Elements
of the
sample space
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Events
Subsets
of the
sample space
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