### 数学代写|概率论代写Probability theory代考|STAT4028

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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

## 数学代写|概率论代写Probability theory代考|Random trial, sample space, and event

Consider a trial with an unknown prior result, yet known possible results. Such a trial is called a random trial, and the set of its possible results is called sample space, usually denoted by the letter $S$. For more clarification, consider the following examples:
In the trial of tossing one coin, the sample space is as follows:
$$S={H, T}^1$$
In the trial of tossing two coins, the sample space is defined as:
$$S={(H, H),(T, H),(H, T),(T, T)}$$

In the trial of tossing two dice, the sample space consists of 36 states and is defined as:
$$S={(i, j): i, j=1,2,3,4,5,6}$$
In the trial of measuring the lifetime of a particular light bulb (in hours), the sample space is defined as:
$$S={x: x \geq 0}$$
Each subset of a sample space with possible outcomes belonging to a trial is called the sample space event.

For instance, consider the trial of tossing two coins. If the event $E$ denotes at least one heads appears, the event is expressed as follows:
$$E={(H, T),(T, H),(H, H)}$$
Alternatively, consider the trial of tossing two dice. If the event $E$ denotes the sum of the results of two dice is equal to 4 , the event is expressed as:
$$E={(1,3),(2,2),(3,1)}$$
Also, in the trial of measuring the lifetime of a particular light bulb, the event $E$ is defined as the lifetime of the light bulb with a maximum value of 10 hours. This event is represented as follows:
$$E={x: 0 \leq x \leq 10}$$
Note that we say the event $E$ has occurred when one of its results has occurred. Namely, in the trial of tossing two dice, assume that the event $E$ denotes the sum of the results of two dice is equal to 4 . Then, if one of the results $(1,3),(2,2)$, or $(3,1)$ occurs, we say that the event $E$ has occurred.

## 数学代写|概率论代写Probability theory代考|An introduction to the algebra of sets

In the probability theory, the algebra of sets and the relationships between different Levents of a trial are of great importance, which are addressed in this section. Meanwhile, we assume that all the studied events belong to one sample space such as S.

One illustrative method to indicate the logical relationships of events is the use of the Venn Diagram. In this diagram, the sample space of the trial is represented by a rectangle containing all the points, and the various events such as $\mathrm{E}$ and $\mathrm{F}$ are usually shown as circles inside the rectangle. Thus, the desired events can be shown by hatching the related area of the figure.

If $\mathrm{E}$ and $\mathrm{F}$ are arbitrary two events of the sample space, then we say that $E \cap F$ or $E F$ is the intersection of two cvents $E$ and $F$. That is, it contains all possible results of the trial, which are both in the events E and F.

In fact, $E \cap F$ occurs whenever both of the events $\mathrm{E}$ and $\mathrm{F}$ occur. For this purpose, a result of the sample space should occur that is in common for both of the events.

Moreover, we say that $E \cup F$ is the union of two events $E$ and $F$ whenever it contains all results either in $\mathrm{E}$ or $\mathrm{F}$ (or both), as shown Figure $2-1$.

In other words, $E \cup F$ occurs whenever at least one of the events $E$ and $F$ occurs. To this end, a result of the sample space should occur that is either in $E$ or $F$ (or both), shown as $E \cup F$.

Namely, in the trial of tossing a die, suppose that the events $E$ and $F$ are defined as $E={1,2,3}$ and $F={3,4}$, respectively. Then, the events $E \cap F$ and $E \cup F$ will lead to the respective values ${3}$ and ${1,2,3,4}$.

# 概率论代考

## 数学代写|概率论代写Probability theory代考|Random trial, sample space, and event

$$S=H, T^1$$

$$S=(H, H),(T, H),(H, T),(T, T)$$

$$S=(i, j): i, j=1,2,3,4,5,6$$

$$S=x: x \geq 0$$

$$E=(H, T),(T, H),(H, H)$$

$$E=(1,3),(2,2),(3,1)$$

$$E=x: 0 \leq x \leq 10$$

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## MATLAB代写

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