概率论与随机过程笔记（1）：样本空间与概率
概率论与随机过程笔记（1）：样本空间与概率
2019-10-27
这部分的笔记依据Dimitri P. Bertkas和John N. Tsitsiklis的第1章内容（不包括1.6节组合数学的内容）。鉴于线性代数的笔记中⼤量latex公式输⼊中，切换中英⽂输⼊法浪费了很多时间，所以概率笔记会⽤英⽂完成。
国际英标1.1 集合（ts ）
【集合的定义】A t is a collection of objects, which are the elements of the t. If is a t and is an element of , we
write . If is not an element of , we write . A t can have no elements, in which ca it is called the empty t,denoted by 【集合的表⽰⽅法】Sets can be specified in a variety of ways:
The symbol “” is to be read as “such that.”
【集合之间的关系】If every element of a t is also an element of a t , we say that is a subt of , and we write or . If and , the two ts are equal, and we write .
【空间 】It is also expedient to introduce a universal t, denoted by , which contains all objects that could conceivably be of interest in a particular context. Having specified the context in terms of a universal t , we only consider ts that are subts of .
【补集】The complement of a t , with respect to the univer , is the t of all the elements of that do not belong to , and is denoted by . Note that .
【集合的交和并】The union of two ts and is the t of all elements that belong to or (or both), and is denoted by . The interction of two ts and is the t of all elements that belong to both and , and is denoted by .Thus,
【不想交 & 分割】Two ts are said to be disjoint if their interction is empty. More generally, veral ts are said to be disjoint if no two of them have a common element. A collection of ts is said to be a partition of a t if the ts in the collection are disjoint and their union is .
S x S x ∈S x S x ∈
/
S ∅
S ={x ,x ,⋯,x }
12n S ={x ,x ,⋯}
12{x ∣x satisfies P }
∣S T S T S ⊂T T ⊃S S ⊂T T ⊂S S =T ΩΩΩS ΩS Ω{x ∈Ω∣x ∈
/S }ΩS S c Ω=c ∅S T S T S ∪T S T S T S ∩T S ∪T ={x ∣x ∈S or x ∈T }
S ∩T ={x ∣x ∈S and x ∈T }
=n =1⋃∞
S ∪1S ⋯=2{x ∣x ∈S for some n }
n =n =1⋂∞S ∩1S ⋯=2{x ∣x ∈S for alle n }
talent是什么意思
n S S
If and are two objects. we u to denote the ordered pair of and . The t of scalars (real numbers) is denoted by : the t of pairs (or triplets) of scalars, i.e … the two-dimensional plane (or three-dimensional space, respectively) is denoted by (or . respectively).
Sets and the associated operations are easy to visualize in terms of Venn diagrams. as illustrated in Fig. 1.1.
【矩阵代数】Set operations have veral properties, which are elementary conquences of the definitions. Some exa1nples are:
fox newsTwo particularly uful properties are given by De Morgan’s laws which state that:
1.2 概率模型（probablistic models ）
x y (x .y )x y R R 2R 3S ∪T =T ∪S
S ∪(T ∪U )=(S ∪T )∪U
S ∩(T ∪U )=(S ∩T )∪(S ∩U )
S ∪(T ∩U )=(S ∪T )∩(S ∪U )
(S )=c c S
S ∩S =c ∅
S ∪Ω=Ω
S ∩Ω=S
(S )=n ⋃n c S n ⋂n c
(S )=n ⋂n c S n ⋃n c
A probabilistic model is a mathematical description of an uncertain situation.Its two main ingredients are listed below and
are visualized in Fig. 1.2.Elements of a Probabilistic Model：
The sample space , which is the t of all possible outcomes of an experiment.
The probability law, which assigns to a t of possible outcomes (also called an event) a nonnegative number (called the probability of ) that encodes our knowledge or belisf about the collective “likelihood” of the elements of .
【样本空间和事件】Every probabilistic model involves an underlying process, called the experiment, that will produce exactly one out of veral possible outcomes. The t of all possible outcomes is called the sample space of the experiment, and is denoted by . A subt of the sample space, that is, a collection of possible outcomes, is called an event. here is no restriction on what constitutes an experiment.
The sample space of an experiment may consist of a finite or an infinite number of possible outcomes. Finite sample spaces are conceptually and mathematically simpler.
Generally, the sample space chon for a probabilistic model must be collectively exhaustive, in the n that no matter what happens in the experiment, we always obtain an outcome that has been included in the sample space. In addition, the sample space should have enough detail to distinguish between all outcomes of interest to the modeler, while avoiding irrelevant details.
【概率律（probability law.）】the probabilitylaw assigns to every event . a number , called the proba
bility of .satisfying the following axioms:
(Nonnegativity) , for every event (Additivity) If and are two disjoint events, then the probability of their union satisfies
. More generally, if the sample space has an infinite number of elements and , , is a quence of disjoint events,then the probability of their union satifies
(Normalization) The probability of the entire sample space is equal to 1, that is, 【离散概率律（Discrete Probability Law）】If the sample space consists of a finite number of possible outcomes, then the probability law is specified by the probabilities of the events that consist of a single element. In particular, the probability of any event is the sum of the probabilities of its elements:猕猴桃的英文
ΩA P (A )A A ΩA P (A )A P (A )≥0A
A B P (A ∪B )=P (A )+P (B )
A 1A 2⋯P (A ∪1A ∪2⋯)=P (A )+1P (A )+2⋯
ΩP (Ω)=1
{s ,s ,⋯,s }12n P ({s ,s ,⋯,s })=12n P (s )+1P (s )+2⋯+P (s )
n
【离散均匀概率律（古典模型）（Discrete Uniform Probability Law）】In the special ca where the probabilities
are all the same (by necessity equal to , in view of the normalization axiom). If the sample space consists of n possible outcomes which are equally likely (i.e, all single-element events have he same probability), then the probability of any event is given by
【连续模型】Probabilistic models with continuous sample spaces differ from their discrete counterparts in that the
probabilities of the single-element events may not be sufficient to characterize the probability law. This is illustrated in the following examples, which also indicate how to generalize the uniform probability law to the ca of a continuous sample space.
A wheel of fortune is continuously calibrated from 0 to 1, so the possible outcomes of an experiment consisting of a single spin are the numbers in the interval . Assuming a fair wheel, it is appropriate to
consider all outcomes equally likely, but what is the probability of the event consisting of a single element? It cannot be positive, becau
then, using the additivity axiom, it would follow that events with a sufficiently large number of elements would have probability larger than 1. Therefore, the probability of any event that consists of a single element must be 0.
In this example, it makes n to assign probability to any subinterval of , and to calculate the
probability of a more complicated t by evaluating its “length”. () This assignment satisfies the three probability axioms and qualifies as a legitimate probability law.
【概率律的性质】Probability laws have a number of properties, which can be deduced from the axioms. Some of them are summarized below.
if , then
.The properties, and other similar ones:
1.3 条件概率（conditional probability ）P (s ),⋯,P (s )1n 1/n A P (A )=n
number of elements of A
n =[0,1]b −a [a ,b ][0,1]dt ∫S A ⊂B P (A )≤P (B )P (A ∪B )=P (A )+P (B )−P (A ∩B )
P (A ∪B )≤P (A )+P (B )
plotterqvodzyP (A ∪B ∪C )=P (A )+P (A ∩c B )+P (A ∩c B ∩c C )
P (A ∪1A ∪2⋯∪A )≤n P (A )
i =1∑n
i P (A ∪1A ∪2⋯∪A )≤n P (A )+1P (A ∪2⋯∪A )
n
Conditional probability provides us with a way to reason about the outcome of an experiment, bad on partial information.In more preci terms, given an experiment, a corresponding sample space, and a probability law, suppo that we know that the outcome is within some given event . We wish to quantify the likelihood that the outcome also belongs to some other given event . We thus ek to construct a new probability law that takes into account the available knowledge: a probability law that for any event A. specifies the conditional probability of given . denoted by .
We would like the conditional probabilities of different events to constitute a legitimate probability law, which satisfies the probability axioms. Generalizing the argument, we introduce the following definition of conditional probability:
In words, out of the total probability of the elements of , is the fraction that is assigned to possible outcomes that also belong to .【条件概率的概率律/性质】
The conditional probability of an event , given an event with , is defined byquiteafew
and specifies a new (conditional) probability law on the same sample space . In particular, all properties of probability laws remain valid for conditional probability laws.
Conditional probabilities can also be viewed as a probability law on a new univer , becau all of the conditional probability is concentrated on .
If the possible outcomes are finitely many and equally likely, then
鹅的英文
【利⽤条件概率定义概率模型】When constructing probabilistic models for experiments that have a quential character, it is often natural and convenient to first specify conditional probabilities and then u them to determine unconditional
probabilities. The rule , which is a restatement of the definition of conditional probability, is often helpful in this process.
秋冬服装搭配
【贯序树形图】We have a general rule for calculating various probabilities in conjunction with a tree-bad quential description of an experiment. In particular:
We t up the tree so that an event of interest is associated with a leaf. We view the occurrence of the event as a quence of steps, namely, the traversals of the branches along the path fron1 the root to the leaf.We record the conditional probabilities associated with the branches of the tree.
We obtain the probability of a leaf by multiplying the probabilities recorded along the corresponding path of the tree.In mathematical terms, we are dealing with an event which occurs if and only if each one of veral events has occurred, i.e., . The occurrence of is viewed as an occurrence of , followed by the occurrence of , then of , etc., and it is visualized as a path with n branches, corresponding to the events . The probability of is given by the following rule:B A A B P (A ∣B )P (A ∣B )A P (A ∣B )=P (B )
P (A ∩B )
B P (A ∣B )A A B P (B )>0P (A ∣B )=P (B )
P (A ∩B )
大学梦ΩB B P (A ∣B )=number of elements of B
number of elements of A ∩B
P (A ∩B )=P (B )P (A ∣B )A A ,⋯,A 1n A =A ∩1A ∩⋅⋅⋅∩A 2n A A 1A 2A 3A ,⋯,A 1n A
