Probability Theory Presentation 01
Transcript of Probability Theory Presentation 01
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BST 401 Probability Theory
Xing Qiu Ha Youn Lee
Department of Biostatistics and Computational BiologyUniversity of Rochester
September 2, 2010
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Outline
1 Set and Functions
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Set Operations
The whole set , the empty set ; an element x in a set A:x A.
A B, A B; A B, A B.
Set operations: A B, A B, Ac
(w.r.t. ).De Morgans laws: (A B)c = AcBc, (A B)c = AcBc.
For more than two sets:
n
Anc
=
nAcn,
n
Anc
=
nAcn.
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Set Operations
The whole set , the empty set ; an element x in a set A:x A.
A B, A B; A B, A B.
Set operations: A B, A B, Ac
(w.r.t. ).De Morgans laws: (A B)c = AcBc, (A B)c = AcBc.
For more than two sets:
n
Anc
=
nAcn,
n
Anc
=
nAcn.
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Set Operations
The whole set , the empty set ; an element x in a set A:x A.
A B, A B; A B, A B.
Set operations: A B, A B, Ac
(w.r.t. ).De Morgans laws: (A B)c = AcBc, (A B)c = AcBc.
For more than two sets:
n
Anc
=
nAcn,
n
Anc
=
nAcn.
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Set Operations
The whole set , the empty set ; an element x in a set A:x A.
A B, A B; A B, A B.
Set operations: A B, A B, Ac
(w.r.t. ).De Morgans laws: (A B)c = AcBc, (A B)c = AcBc.
For more than two sets:
n
Anc
=
nAcn,
n
Anc
=
nAcn.
Qiu, Lee BST 401
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Set Operations
The whole set , the empty set ; an element x in a set A:x A.
A B, A B; A B, A B.
Set operations: A B, A B, Ac
(w.r.t. ).De Morgans laws: (A B)c = AcBc, (A B)c = AcBc.
For more than two sets:
n
Anc
=
nAcn,
n
Anc
=
nAcn.
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Disjoint atoms: the classical three-circle-diagram.
Figure: Three sets can generate 7 atoms.
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Functions
Please review the basic definitions of a function. Pay
attention to a functions domain and its image.
I assume you know the definition and basic properties ofthe following elementary functions:
1 Power functions, xa, a R. When a is a non-integerrational number, without confusion we take the principlebranch of nth root operation; when a is irrational, itsdefinition is given by the principle branch of a log function.
2 Exponential and logarithmic functions, ex and log(x).3
Trigonometric functions and their inverse functions. Theirdomain, image, etc.4 The combination of the above by +,,, and functional
composition.
Qiu, Lee BST 401
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Functions
Please review the basic definitions of a function. Pay
attention to a functions domain and its image.
I assume you know the definition and basic properties ofthe following elementary functions:
1 Power functions, xa, a R. When a is a non-integerrational number, without confusion we take the principlebranch of nth root operation; when a is irrational, itsdefinition is given by the principle branch of a log function.
2 Exponential and logarithmic functions, ex and log(x).3
Trigonometric functions and their inverse functions. Theirdomain, image, etc.4 The combination of the above by +,,, and functional
composition.
Qiu, Lee BST 401
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Functions
Please review the basic definitions of a function. Pay
attention to a functions domain and its image.
I assume you know the definition and basic properties ofthe following elementary functions:
1 Power functions, xa, a R. When a is a non-integerrational number, without confusion we take the principlebranch of nth root operation; when a is irrational, itsdefinition is given by the principle branch of a log function.
2 Exponential and logarithmic functions, ex and log(x).3
Trigonometric functions and their inverse functions. Theirdomain, image, etc.4 The combination of the above by +,,, and functional
composition.
Qiu, Lee BST 401
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Functions
Please review the basic definitions of a function. Pay
attention to a functions domain and its image.
I assume you know the definition and basic properties ofthe following elementary functions:
1 Power functions, xa, a R. When a is a non-integerrational number, without confusion we take the principlebranch of nth root operation; when a is irrational, itsdefinition is given by the principle branch of a log function.
2 Exponential and logarithmic functions, ex and log(x).3
Trigonometric functions and their inverse functions. Theirdomain, image, etc.4 The combination of the above by +,,, and functional
composition.
Qiu, Lee BST 401
http://find/http://goback/ -
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Functions
Please review the basic definitions of a function. Pay
attention to a functions domain and its image.
I assume you know the definition and basic properties ofthe following elementary functions:
1 Power functions, xa, a R. When a is a non-integerrational number, without confusion we take the principlebranch of nth root operation; when a is irrational, itsdefinition is given by the principle branch of a log function.
2 Exponential and logarithmic functions, ex and log(x).3
Trigonometric functions and their inverse functions. Theirdomain, image, etc.4 The combination of the above by +,,, and functional
composition.
Qiu, Lee BST 401
http://find/http://goback/ -
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Functions
Please review the basic definitions of a function. Pay
attention to a functions domain and its image.
I assume you know the definition and basic properties ofthe following elementary functions:
1 Power functions, xa, a R. When a is a non-integerrational number, without confusion we take the principlebranch of nth root operation; when a is irrational, itsdefinition is given by the principle branch of a log function.
2 Exponential and logarithmic functions, ex and log(x).3
Trigonometric functions and their inverse functions. Theirdomain, image, etc.4 The combination of the above by +,,, and functional
composition.
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Limit of a sequence of real numbers (I)
For simplicity, we are going to use increasing to meannon-decreasing. decreasing to mean non-increasing.
A sequence of real numbers (ai) = (a1, a2, . . .) convergesto a if for any given precision criterion > 0, there exists
an integer N such that the error, defined as
dist(ai a) = |ai a
|, is smaller than for all i N.
An increasing, bounded sequence of real numbers
a1 a2 . . . always converges to a limit. (if you consider as a valid limiting point, the boundedness part can be
omitted.)Similarly, a decreasing sequence of real numbers always
converges to a limit (if you dont like , you can add thebounded from below condition).
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Limit of a sequence of real numbers (I)
For simplicity, we are going to use increasing to meannon-decreasing. decreasing to mean non-increasing.
A sequence of real numbers (ai) = (a1, a2, . . .) convergesto a if for any given precision criterion > 0, there exists
an integer N such that the error, defined as
dist(ai a) = |ai a
|, is smaller than for all i N.
An increasing, bounded sequence of real numbers
a1 a2 . . . always converges to a limit. (if you consider as a valid limiting point, the boundedness part can be
omitted.)Similarly, a decreasing sequence of real numbers always
converges to a limit (if you dont like , you can add thebounded from below condition).
Qiu, Lee BST 401
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Limit of a sequence of real numbers (I)
For simplicity, we are going to use increasing to meannon-decreasing. decreasing to mean non-increasing.
A sequence of real numbers (ai) = (a1, a2, . . .) convergesto a if for any given precision criterion > 0, there exists
an integer N such that the error, defined as
dist(ai a) = |ai a
|, is smaller than for all i N.
An increasing, bounded sequence of real numbers
a1 a2 . . . always converges to a limit. (if you consider as a valid limiting point, the boundedness part can be
omitted.)Similarly, a decreasing sequence of real numbers always
converges to a limit (if you dont like , you can add thebounded from below condition).
Qiu, Lee BST 401
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Limit of a sequence of real numbers (I)
For simplicity, we are going to use increasing to meannon-decreasing. decreasing to mean non-increasing.
A sequence of real numbers (ai) = (a1, a2, . . .) convergesto a if for any given precision criterion > 0, there exists
an integer N such that the error, defined as
dist(ai a) = |ai a
|, is smaller than for all i N.
An increasing, bounded sequence of real numbers
a1 a2 . . . always converges to a limit. (if you consider as a valid limiting point, the boundedness part can be
omitted.)Similarly, a decreasing sequence of real numbers always
converges to a limit (if you dont like , you can add thebounded from below condition).
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Limit of a sequence of real numbers (II)
In general a sequence of real numbers (a1, a2, . . .) alwayshas a subsequence which approaches the upper limit
(including as a possible limit) of this sequence.
Similarly, it contains a subsequence which approaches its
lower limit. Once the upper limit equals the lower limit, we
say this sequence converges to this limit.
Two companion subsequences, denoted as (b1, b2, . . .)and (c1, c2, . . .), can be quite useful:
bi = supin ai, ci = infinai.
(bi) is decreasing and (ci) is increasing and they convergeto lim supi ai and lim infi ai, respectively.
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Limit of a sequence of real numbers (II)
In general a sequence of real numbers (a1, a2, . . .) alwayshas a subsequence which approaches the upper limit
(including as a possible limit) of this sequence.
Similarly, it contains a subsequence which approaches its
lower limit. Once the upper limit equals the lower limit, we
say this sequence converges to this limit.
Two companion subsequences, denoted as (b1, b2, . . .)and (c1, c2, . . .), can be quite useful:
bi = supin ai, ci = infinai.
(bi) is decreasing and (ci) is increasing and they convergeto lim supi ai and lim infi ai, respectively.
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Limit of a sequence of real numbers (II)
In general a sequence of real numbers (a1, a2, . . .) alwayshas a subsequence which approaches the upper limit
(including as a possible limit) of this sequence.
Similarly, it contains a subsequence which approaches its
lower limit. Once the upper limit equals the lower limit, we
say this sequence converges to this limit.
Two companion subsequences, denoted as (b1, b2, . . .)and (c1, c2, . . .), can be quite useful:
bi = supin ai, ci = infinai.
(bi) is decreasing and (ci) is increasing and they convergeto lim supi ai and lim infi ai, respectively.
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Limit of a sequence of real numbers (II)
In general a sequence of real numbers (a1, a2, . . .) alwayshas a subsequence which approaches the upper limit
(including as a possible limit) of this sequence.
Similarly, it contains a subsequence which approaches its
lower limit. Once the upper limit equals the lower limit, we
say this sequence converges to this limit.
Two companion subsequences, denoted as (b1, b2, . . .)and (c1, c2, . . .), can be quite useful:
bi = supin ai, ci = infinai.
(bi) is decreasing and (ci) is increasing and they convergeto lim supi ai and lim infi ai, respectively.
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Limit of a sequence of real numbers (IV)
For a sequence (an), If its upper limit equals its lower limit
(lim supnan = lim infnan = a), then (an) converges to a.The distance function which quantifies error is important.
For real numbers, there is essentially one way to measure
the error term: |ai a|. This is because a distance
function needs to satisfy several axioms (use wikipedia).
For a sequence of n-dimensional points(vectors), the
natural way to measure the error term is the Euclidean
distance. But other distance functions do exist, such as the
Manhattan distance (google it). Fortunately, a sequence of
vectors is convergent in one distance implies it isconvergent in all other distances.
Unfortunately, you can define quite a few non-compatible
distances of random numbers. So there are many different
convergences of random numbers.
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Limit of a sequence of real numbers (IV)
For a sequence (an), If its upper limit equals its lower limit
(lim supnan = lim infnan = a), then (an) converges to a.The distance function which quantifies error is important.
For real numbers, there is essentially one way to measure
the error term: |ai a|. This is because a distance
function needs to satisfy several axioms (use wikipedia).
For a sequence of n-dimensional points(vectors), the
natural way to measure the error term is the Euclidean
distance. But other distance functions do exist, such as the
Manhattan distance (google it). Fortunately, a sequence of
vectors is convergent in one distance implies it isconvergent in all other distances.
Unfortunately, you can define quite a few non-compatible
distances of random numbers. So there are many different
convergences of random numbers.
Qiu, Lee BST 401
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Limit of a sequence of real numbers (IV)
For a sequence (an), If its upper limit equals its lower limit
(lim supnan = lim infnan = a), then (an) converges to a.The distance function which quantifies error is important.
For real numbers, there is essentially one way to measure
the error term: |ai a|. This is because a distance
function needs to satisfy several axioms (use wikipedia).
For a sequence of n-dimensional points(vectors), the
natural way to measure the error term is the Euclidean
distance. But other distance functions do exist, such as the
Manhattan distance (google it). Fortunately, a sequence of
vectors is convergent in one distance implies it isconvergent in all other distances.
Unfortunately, you can define quite a few non-compatible
distances of random numbers. So there are many different
convergences of random numbers.
Qiu, Lee BST 401
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Limit of a sequence of real numbers (IV)
For a sequence (an), If its upper limit equals its lower limit
(lim supnan = lim infnan = a), then (an) converges to a.The distance function which quantifies error is important.
For real numbers, there is essentially one way to measure
the error term: |ai a|. This is because a distance
function needs to satisfy several axioms (use wikipedia).
For a sequence of n-dimensional points(vectors), the
natural way to measure the error term is the Euclidean
distance. But other distance functions do exist, such as the
Manhattan distance (google it). Fortunately, a sequence of
vectors is convergent in one distance implies it isconvergent in all other distances.
Unfortunately, you can define quite a few non-compatible
distances of random numbers. So there are many different
convergences of random numbers.
Qiu, Lee BST 401
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Limit of a function
We can now define the limit of a function f(x) as xapproaches x0. x0 could be , for the sake of simplicitywe assume x0 is finite for the following definition.
limxx0 f(x) = y if and only if
> 0, > 0 such that dist(f(x)y
) < for all x Ball(x0, ).
The logic negation of a sequence/function converges to a
value is that this sequence/function breaks the precision
rule infinitely often. More precisely, a sequence is not
convergent if for a given > 0, we have
dist(ai, a) > i.o.
where i.o. stands for infinitely often.
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Limit of a function
We can now define the limit of a function f(x) as xapproaches x0. x0 could be , for the sake of simplicitywe assume x0 is finite for the following definition.
limxx0 f(x) = y if and only if
> 0, > 0 such that dist(f(x)y
) < for all x Ball(x0, ).
The logic negation of a sequence/function converges to a
value is that this sequence/function breaks the precision
rule infinitely often. More precisely, a sequence is not
convergent if for a given > 0, we have
dist(ai, a) > i.o.
where i.o. stands for infinitely often.
Qiu, Lee BST 401
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Limit of a function
We can now define the limit of a function f(x) as xapproaches x0. x0 could be , for the sake of simplicitywe assume x0 is finite for the following definition.
limxx0 f(x) = y if and only if
> 0, > 0 such that dist(f(x)y
) < for all x Ball(x0, ).
The logic negation of a sequence/function converges to a
value is that this sequence/function breaks the precision
rule infinitely often. More precisely, a sequence is not
convergent if for a given > 0, we have
dist(ai, a) > i.o.
where i.o. stands for infinitely often.
Qiu, Lee BST 401
http://find/http://goback/