### 统计代写|蒙特卡洛方法代写monte carlo method代考|PRELIMINARIES

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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|蒙特卡洛方法代写monte carlo method代考|RANDOM EXPERIMENTS

The basic notion in probability theory is that of a random experiment: an experiment whose outcome cannot be determined in advance. The most fundamental example is the experiment where a fair coin is tossed a number of times. For simplicity suppose that the coin is tossed three times. The sample space, denoted $\Omega$, is the set of all possible outcomes of the experiment. In this case $\Omega$ has eight possible outcomes:
$$\Omega={H H H, H H T, H T H, H T T, T H H, T H T, T T H, T T T},$$
where, for example, HTH means that the first toss is heads, the second tails, and the third heads.

Subsets of the sample space are called events. For example, the event $A$ that the third toss is heads is
$$A={H H H, H T H, T H H, T T H} .$$
We say that event $A$ occurs if the outcome of the experiment is one of the elements in $A$. Since events are sets, we can apply the usual set operations to them. For example, the event $A \cup B$, called the union of $A$ and $B$, is the event that $A$ or $B$ or both occur, and the event $A \cap B$, called the intersection of $A$ and $B$, is the event that $A$ and $B$ both occur. Similar notation holds for unions and intersections of more than two events. The event $A^{c}$, called the complement of $A$, is the event that $A$ does not occur. Two events $A$ and $B$ that have no outcomes in common, that is, their intersection is empty, are called disjoint events. The main step is to specify the probability of each event.

Definition 1.2.1 (Probability) A probability $\mathbb{P}$ is a rule that assigns a number $0 \leqslant \mathbb{P}(A) \leqslant 1$ to each event $A$, such that $\mathbb{P}(\Omega)=1$, and such that for any sequence $A_{1}, A_{2}, \ldots$ of disjoint events
$$\mathbb{P}\left(\bigcup_{i} A_{i}\right)=\sum_{i} \mathbb{P}\left(A_{i}\right)$$
Equation (1.1) is referred to as the sum rule of probability. It states that if an event can happen in a number of different ways, but not simultaneously, the probability of that event is simply the sum of the probabilities of the comprising events.

For the fair coin toss experiment the probability of any event is easily given. Namely, because the coin is fair, each of the eight possible outcomes is equally likely, so that $\mathbb{P}({H H H})=\cdots=\mathbb{P}({T T T})=1 / 8$. Since any event $A$ is the union of the “elementary” events ${H H H}, \ldots,{T T T}$, the sum rule implies that
$$\mathbb{P}(A)=\frac{|A|}{|\Omega|},$$
where $|A|$ denotes the number of outcomes in $A$ and $|\Omega|=8$. More generally, if a random experiment has finitely many and equally likely outcomes, the probability is always of the form (1.2). In that case the calculation of probabilities reduces to counting.

## 统计代写|蒙特卡洛方法代写monte carlo method代考|CONDITIONAL PROBABILITY AND INDEPENDENCE

How do probabilities change when we know that some event $B \subset \Omega$ has occurred? Given that the outcome lies in $B$, the event $A$ will occur if and only if $A \cap B$ occurs, and the relative chance of $A$ occurring is therefore $\mathbb{P}(A \cap B) / \mathbb{P}(B)$. This leads to the definition of the conditional probability of $A$ given $B$ :
$$\mathbb{P}(A \mid B)=\frac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)} .$$
For example, suppose that we toss a fair coin three times. Let $B$ be the event that the total number of heads is two. The conditional probability of the event $A$ that the first toss is heads, given that $B$ occurs, is $(2 / 8) /(3 / 8)=2 / 3$.

Rewriting (1.3) and interchanging the role of $A$ and $B$ gives the relation $\mathbb{P}(A \cap$ $B)=\mathbb{P}(A) \mathbb{P}(B \mid A)$. This can be generalized easily to the product rule of probability, which states that for any sequence of events $A_{1}, A_{2}, \ldots, A_{n}$,
$$\mathbb{P}\left(A_{1} \cdots A_{n}\right)=\mathbb{P}\left(A_{1}\right) \mathbb{P}\left(A_{2} \mid A_{1}\right) \mathbb{P}\left(A_{3} \mid A_{1} A_{2}\right) \cdots \mathbb{P}\left(A_{n} \mid A_{1} \cdots A_{n-1}\right)$$
using the abbreviation $A_{1} A_{2} \cdots A_{k} \equiv A_{1} \cap A_{2} \cap \cdots \cap A_{k}$.
Suppose that $B_{1}, B_{2}, \ldots, B_{n}$ is a partition of $\Omega$. That is, $B_{1}, B_{2}, \ldots, B_{n}$ are disjoint and their union is $\Omega$. Then, by the sum rule, $\mathbb{P}(A)=\sum_{i=1}^{n} \mathbb{P}\left(A \cap B_{i}\right)$ and hence, by the definition of conditional probability, we have the law of total probability:
$$\mathbb{P}(A)=\sum_{i=1}^{n} \mathbb{P}\left(A \mid B_{i}\right) \mathbb{P}\left(B_{i}\right)$$
Combining this with the definition of conditional probability gives Bayes’ rule:
$$\mathbb{P}\left(B_{j} \mid A\right)=\frac{\mathbb{P}\left(A \mid B_{j}\right) \mathbb{P}\left(B_{j}\right)}{\sum_{i=1}^{n} \mathbb{P}\left(A \mid B_{i}\right) \mathbb{P}\left(B_{i}\right)}$$
Independence is of crucial importance in probability and statistics. Loosely speaking, it models the lack of information between events. Two events $A$ and $B$ are said to be independent if the knowledge that $B$ has occurred does not change the probability that $A$ occurs. That is, $A, B$ independent $\Leftrightarrow \mathbb{P}(A \mid B)=\mathbb{P}(A)$. Since $\mathbb{P}(A \mid B)=\mathbb{P}(A \cap B) / \mathbb{P}(B)$, an alternative definition of independence is
$A, B$ independent $\Leftrightarrow \mathbb{P}(A \cap B)=\mathbb{P}(A) \mathbb{P}(B)$.
This definition covers the case where $B=\emptyset$ (empty set). We can extend this definition to arbitrarily many events.

## 统计代写|蒙特卡洛方法代写monte carlo method代考|RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

Specifying a model for a random experiment via a complete description of $\Omega$ and $\mathbb{P}$ may not always be convenient or necessary. In practice, we are only interested in certain observations (i.e., numerical measurements) in the experiment. We incorporate these into our modeling process via the introduction of random variables, usually denoted by capital letters from the last part of the alphabet (e.g., $X$, $\left.X_{1}, X_{2}, \ldots, Y, Z\right)$.
EXAMPLE 1.2
We toss a biased coin $n$ times, with $p$ the probability of heads. Suppose that we are interested only in the number of heads, say $X$. Note that $X$ can tale any of the values in ${0,1, \ldots, n}$. The probability distribution of $X$ is given by the binomial formula
$$\mathbb{P}(X=k)=\left(\begin{array}{l} n \ k \end{array}\right) p^{k}(1-p)^{n-k}, \quad k=0,1, \ldots, n$$
Namely, by Example $1.1$, each elementary event ${H T H \cdots T}$ with exactly $k$ heads and $n-k$ tails has probability $p^{k}(1-p)^{n-k}$, and there are $\left(\begin{array}{l}n \ k\end{array}\right)$ such events.

The probability distribution of a general random variable $X$ – identifying such probabilities as $\mathbb{P}(X=x), \mathbb{P}(a \leqslant X \leqslant b)$, and so on – is completely specified by the cumulative distribution function (cdf), defined by
$$F(x)=\mathbb{P}(X \leqslant x), \quad x \in \mathbb{R} .$$
A random variable $X$ is said to have a discrete distribution if, for some finite or countable set of values $x_{1}, x_{2}, \ldots, \mathbb{P}\left(X=x_{i}\right)>0, i=1,2, \ldots$ and $\sum_{i} \mathbb{P}\left(X=x_{i}\right)=$ 1. The function $f(x)=\mathbb{P}(X=x)$ is called the probability mass function (pmf) of $X$ – but see Remark 1.4.1.

## 统计代写|蒙特卡洛方法代写monte carlo method代考|RANDOM EXPERIMENTS

Ω=HHH,HH,HH,H,HH,H,H,,

|一种|一种|Ω|=8

## 统计代写|蒙特卡洛方法代写monte carlo method代考|RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

1.1HHķnķpķ(1−p)nķ(n ķ)

F(X)=磷(X⩽X),X∈R.
XX1,X2,…,磷(X=X一世)>0,一世=1,2,…∑一世磷(X=X一世)=F(X)=磷(X=X)X

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