### 统计代写|概率论作业代写Probability and Statistics代考5CCM241A|Axioms of the probability theory

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|概率论作业代写Probability and Statistics代考5CCM241A|General probability space

In Chapter I we considered the discrete space of elementary events and introduced the notion of a discrete probability space (Chapter I, $\S 2$ ). In it, an event is a subset of the discrete space of elementary events $\Omega=\left{\omega_{1}, \omega_{2}, \ldots\right}$ and the probability of an event $A$
$$A \in \mathcal{A}={A: A \subseteq \Omega}$$
is defined as the sum of the probabilities of all elementary events $\omega \in A$ leading to the event $A$, i.e.
$$P(A)=\sum_{\omega \in A} P(\omega) .$$
After that, a classical definition of probability was given and a number of probability properties derived from this definition were given. For example, it was proved that the probability (probability function) $P$ on $\mathcal{A}$ has the following properties:
1) For any $A \in \mathcal{A}, P(A) \geq 0$;
2) $P(\Omega)=1$;
3) If $A_{1}, A_{2}, \ldots, A_{k}$ are pairwise disjoint events $\left(A_{i} A_{j}=\varnothing, i \neq j\right)$, then
$$P\left(\sum_{i=1}^{k} A_{i}\right)=\sum_{i=1}^{k} P\left(A_{i}\right) .$$

## 统计代写|概率论作业代写Probability and Statistics代考5CCM241A|The necessity of expanding the concept of space elementary events

As we already noted in the first chapter, the space of elementary events $\Omega={\omega}$ corresponding to the experiment under consideration is not necessarily a discrete space of elementary events (that is, a finite or countable set). For example, random throwing

of a point into a segment $t_{1}, t_{2}$ (say, an experiment with temperature measurement) has a continuum of outcomes, because the result may be any point of a segment. If in the experiments that have a finite or countable set of outcomes, any set of outcomes (any subset of the space of elementary events) is an event, then in the example under consideration the situation is different. We will have great difficulties if we consider any subset of this interval as an event.

In order to understand the essence of these difficulties, let us consider the question of constructing a probabilistic model of an experiment consisting of an infinite «independent» coin tossing with the probability of a «Tail» falling out at each step equal to $p$.
As the set of all outcomes (the space of elementary events), it is natural to take the set
$$\Omega=\left{\omega: \omega=\left(\omega_{1}, \omega_{2}, \ldots, \omega_{n}\right): \omega_{i}=0,1\right}$$
where $\omega_{i}$ is a result of the $i$-th trial: if in the $i$-th coin tossing «Tail»s occurs, then $\omega_{i}=1$; if «Head» occurs, then $\omega_{i}=0(i=1,2, \ldots, n)$.
Let us now answer the question: what is the cardinality of the set $\Omega$ ?
First of all, let us recall a well-known result: any number $a \in[0,1)$ can be uniquely decomposed into a set (containing an infinite number of zeros) of binary fractions:
$$a=\frac{a_{1}}{2}+\frac{a_{2}}{2^{2}}+\frac{a_{3}}{2^{3}}+\ldots \quad\left(a_{i}=0 \quad \text { or } \quad a_{i}=1 ; i=1,2,3, \ldots\right)$$
Whence, if we put a number $a=\frac{\omega_{1}}{2}+\frac{\omega_{2}}{2^{2}}+\frac{\omega_{3}}{2^{3}}+\ldots \in[0,1)$ in correspondence to the point (outcome) $\omega=\left(\omega_{1}, \omega_{2}, \omega_{3}, \ldots\right) \in \Omega$, then we see that there is a one-to-one correspondence between the set $\Omega$ and the interval $[0,1)$, and therefore the set has the cardinality of the continuum.

Now, to understand how to define the probability in the introduced model of an infinite number of «independents tossing of the «right» (symmetric) coin, we note the following:

Since it is possible to take as $\Omega$ the set $[0,1)$, then the problem of interest can be considered as a problem of the values of probabilities in the model of a random «choice of a point from a set $[0,1) »$.

From considerations of symmetry it is clear that all outcomes, i.e. all points of the interval $[0,1)$ must be «equally likely». But the set $[0,1)$ is uncountable, and if we assume that its probability is 1 , then the probability $P(\omega)$ of each elementary event $\omega \in[0,1)$ must necessarily be zero.

## 统计代写|概率论作业代写Probability and Statistics代考5CCM241A|Probability in a measurable space

In this subsection we introduce the concept of probability of an event and prove a number of important properties of probability.

Definition 3. Let a measurable space $(\Omega, \mathcal{A})$, where $\Omega={\omega}$ is a sample space, $\mathcal{A}$ is algebra of subsets of $\Omega$, be given.

Then defined on $(\Omega, \mathcal{A})$ probability (probabilistic measure, probabilistic function) is called a numerical function $P$, which is defined on $\mathcal{A}$ and assigns to the event $A \in \mathcal{A}$ its probability $P(A)$ with the following properties:
P1. For any event $A \in \mathcal{A}$ the probability $P(A) \geq 0$;
P2. $P(\Omega)=1$;
P3. If $A_{1}, A_{2}, \ldots \in \mathcal{A}$ is a sequence of pairwise disjoint events $\left(A_{i} A_{j}=\varnothing, i \neq j\right)$ and $\sum_{n=1}^{\infty} A_{n} \in \mathcal{A}$, then
$$P\left(\sum_{n=1}^{\infty} A_{n}\right)=\sum_{n=1}^{\infty} P\left(A_{n}\right)$$
If $\mathcal{A}$ is $\sigma$-algebra, then in the definition 3 the requirement $\sum_{n=1}^{\infty} A_{n} \in \mathcal{A}$ is superfluous (by the definition of $\sigma$-algebra it is automatically satisfied).

In the probability theory, axiom P1 is called the axiom (or property) of nonnegativity, axiom $\mathbf{P 2}$ – axiom (or property) of normalization, axiom $\mathbf{P 3}$ – axiom (or property) of countable additivity or sigma $(\sigma) \sigma$-additivity.

The triple $(\Omega, \mathcal{A}, P)$, where $\mathcal{A}$ is algebra, is called extended probabilistic space.
The triple $(\Omega, \mathcal{F}, P)$, where $\mathcal{F}$ is $\sigma$-algebra, is called a (general) probabilistic space.

If $\Omega$ is a discrete sample space, i.e. a finite or countable set then it is obvious that a system (set) of all subsets is a sigma-algebra and the corresponding triple $(\Omega, \mathcal{F}, P)$ is called a discrete probability space (see $\mathrm{Ch} . \mathrm{I}, \S 1)$. In particular case, when $\Omega$ is a finite set $(|\Omega|<\infty)$, then the triple $(\Omega, \mathcal{F}, P)$ is called a finite probability space.
The construction of a probability space $(\Omega, \mathcal{F}, P)$ is the main stage in the process of constructing a mathematical model of the experiment.

## 统计代写|概率论作业代写Probability and Statistics代考5CCM241A|General probability space

1）对于任何一种∈一种,磷(一种)≥0;
2) 磷(Ω)=1;
3) 如果一种1,一种2,…,一种ķ是成对的不相交事件(一种一世一种j=∅,一世≠j)， 然后

## 统计代写|概率论作业代写Probability and Statistics代考5CCM241A|The necessity of expanding the concept of space elementary events

\Omega=\left{\omega: \omega=\left(\omega_{1}, \omega_{2}, \ldots, \omega_{n}\right): \omega_{i}=0,1\right }\Omega=\left{\omega: \omega=\left(\omega_{1}, \omega_{2}, \ldots, \omega_{n}\right): \omega_{i}=0,1\right }

## 统计代写|概率论作业代写Probability and Statistics代考5CCM241A|Probability in a measurable space

P1。对于任何事件一种∈一种概率磷(一种)≥0;
P2。磷(Ω)=1;
P3。如果一种1,一种2,…∈一种是成对不相交事件的序列(一种一世一种j=∅,一世≠j)和∑n=1∞一种n∈一种， 然后

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