### 统计代写|贝叶斯分析代写Bayesian Analysis代考|STAT4102

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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

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|Probability Notation Where There Are Different

Consider the experiment of rolling two fair dice. There are many different outcomes of interest for this experiment including the following:

• The sum of the two dice rolled (let’s call this outcome $X$ ).
The highest number die rolled (let’s call this outcome $Y$ ).
These two different outcomes of interest have different sets of elementary events.
Outcome $X$ has eleven elementary events: 2,3,4,5,6,7,8,9,10, 11, 12 .
• Outcome $Y$ has six elementary events: 1, 2, 3, 4, 5, 6 .
If we are not careful about specifying the particular outcome of interest for the experiment, then there is the potential to introduce genuine ambiguity when calculating probabilities.

For example, consider the elementary event ” 2 .” What is the probability of observing this event for this experiment? In other words what is $P(2)$ ? The answer depends on whether we are considering outcome $X$ or outcome $Y$ :

• For outcome $X$, the probability $P(2)$ is $1 / 36$ because there are 36 different ways to roll two dice and only one of these, the roll $(1,1)$, results in the sum of the dice being 2 .
• For outcome $Y$, the probability $P(2)$ is $1 / 12$ because of the 36 different ways to roll two dice there are three ways, the rolls $(1,2),(2,1)$ and $(2,2)$, that result in the highest number rolled being 2 .
Because of this ambiguity it is common practice, when there are different outcomes of interest for the same experiment to include some notation that identifies the particular outcome of interest when writing down probabilities. Typically, we would write $P(X=2)$ or $P(Y=2)$ instead of just $P(2)$.

The notation extends to events that comprise more than one elementary event. For example, consider the event $E$ defined as “greater than 3”:

• For outcome $X$, the event is $E$ is equal to ${4,5,6,7,8,9,10,11,12}$.
• For outcome $Y$, the event is $E$ is equal to ${4,5,6}$.
We calculate the probabilities as
• For event $X, P(E)=11 / 12$.
• For event $Y, P(E)=3 / 4$.
Typically we would write $P(X=E)$ or $P(X \geq 3)$ for the former and $P(Y=E)$ or $P(Y \geq 3)$ for the latter.
In this example the outcomes $X$ and $Y$ can be considered as variables whose possible values are their respective set of elementary events. In general, if there is not an obviously unique outcome of interest for an experiment, then we need to specify each outcome of interest as a named variable and include this name in any relevant probability statement.

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|Probability Distributions

Consider the experiment of selecting a contractor to complete a piece of work for you. We are interested in the outcome “quality of the contractor.” Since, as discussed in Box 5.5, this is just one of many possible outcomes of interest for this experiment (others might be price of contractor, experience of contractor, etc.) it is safest to associate a variable name, say $Q$, with the outcome “quality of the contractor.” Let us assume that the set of elementary events for $Q$ is {very poôr, poōr, averāge, good, very good}.

On the basis of our previous experience with contractors, or purely based on subjective judgment, we might assign the probabilities to these elementary events for $Q$ as shown in the table of Figure 5.2(a). Since the numbers are all between 0 and 1 , and since they sum to 1 , this assignment is a valid probability measure for $Q$ (i.e., for the experiment with outcome $Q$ ) because it satisfies the axioms.

A table like the one in Figure 5.2(a), or equivalent graphical representations like the ones in Figure 5.2(b) and Figure 5.2(c), is called a probability distribution. In general, for experiments with a discrete set of elementary events:There is a very common but somewhat unfortunate notation for probability distributions. The probability distribution for an outcome such as $Q$ of an experiment is often written in shorthand as simply: $P(Q)$. If there was an event referred to as $Q$ then the expression $P(Q)$ is ambiguous since it refers to two very different concepts. Generally it will be clear from the context whether $P(Q)$ refers to the probability distribution of an outcome $Q$ or whether it refers to the probability of an event $Q$.

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|Probability Notation Where There Are Different

• 掷出的两个骰子的总和（我们称这个结果为X）。
掷出的最高点数（我们称这个结果为是的）。
这两种不同的兴趣结果具有不同的基本事件集。
结果X有十一个基本事件：2,3,4,5,6,7,8,9,10,11,12。
• 结果是的有六个基本事件：1、2、3、4、5、6。
如果我们在指定实验感兴趣的特定结果时不小心，那么在计算概率时就有可能引入真正的歧义。

• 对于结果X, 概率磷(2)是1/36因为掷两个骰子有 36 种不同的方法，而其中只有一种，掷骰子(1,1)，结果骰子的总和为 2 。
• 对于结果是的, 概率磷(2)是1/12因为掷两个骰子有 36 种不同的方式，所以有三种方式，掷骰子(1,2),(2,1)和(2,2)，这导致滚动的最高数字为 2 。
由于这种模糊性，通常的做法是，当同一实验有不同的感兴趣结果时，在写下概率时包含一些标识感兴趣的特定结果的符号。通常，我们会写磷(X=2)或者磷(是的=2)而不仅仅是磷(2).

• 对于结果X, 事件是和等于4,5,6,7,8,9,10,11,12.
• 对于结果是的, 事件是和等于4,5,6.
我们计算概率为
• 活动X,磷(和)=11/12.
• 活动是的,磷(和)=3/4.
通常我们会写磷(X=和)或者磷(X≥3)对于前者和磷(是的=和)或者磷(是的≥3)对于后者。
在这个例子中，结果X和是的可以被认为是变量，其可能值是它们各自的基本事件集。一般来说，如果一个实验没有明显独特的感兴趣的结果，那么我们需要将每个感兴趣的结果指定为一个命名变量，并将这个名称包含在任何相关的概率陈述中。

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## MATLAB代写

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