### 统计代写|蒙特卡洛方法代写monte carlo method代考|SOME IMPORTANT DISTRIBUTIONS

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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|SOME IMPORTANT DISTRIBUTIONS

Tables $1.1$ and $1.2$ list a number of important continuous and discrete distributions. We will use the notation $X \sim f, X \sim F$, or $X \sim$ Dist to signify that $X$ has a pdf $f$, a cdf $F$ or a distribution Dist. We sometimes write $f_{X}$ instead of $f$ to stress that the pdf refers to the random variable $X$. Note that in Table $1.1, \Gamma$ is the gamma function: $\Gamma(\alpha)=\int_{0}^{\infty} \mathrm{e}^{-x} x^{\alpha-1} \mathrm{~d} x, \quad \alpha>0$

It is often useful to consider different kinds of numerical characteristics of a random variable. One such quantity is the expectation, which measures the mean value of the distribution.

Definition 1.6.1 (Expectation) Let $X$ be a random variable with pdf $f$. The expectation (or expected value or mean) of $X$, denoted by $\mathbb{E}[X]$ (or sometimes $\mu$ ), is defined by
$$\mathbb{E}[X]= \begin{cases}\sum_{x} x f(x) & \text { discrete case } \ \int_{-\infty}^{\infty} x f(x) \mathrm{d} x & \text { continuous case }\end{cases}$$
If $X$ is a random variable, then a function of $X$, such as $X^{2}$ or $\sin (X)$, is again a random variable. Moreover, the expected value of a function of $X$ is simply a weighted average of the possible values that this function can take. That is, for any real function $h$
$$\mathbb{E}[h(X)]= \begin{cases}\sum_{x} h(x) f(x) & \text { discrete case } \ \int_{-\infty}^{\infty} h(x) f(x) \mathrm{d} x & \text { continuous case. }\end{cases}$$
Another useful quantity is the variance, which measures the spread or dispersion of the distribution.

Definition 1.6.2 (Variance) The variance of a random variable $X$, denoted by $\operatorname{Var}(X)$ (or sometimes $\sigma^{2}$ ), is defined by
$$\operatorname{Var}(X)=\mathbb{E}\left[(X-\mathbb{E}[X])^{2}\right]=\mathbb{E}\left[X^{2}\right]-(\mathbb{E}[X])^{2}$$
The square root of the variance is called the standard deviation. Table $1.3$ lists the expectations and variances for some well-known distributions.

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

Often a random experiment is described by more than one random variable. The theory for multiple random variables is similar to that for a single random variable.
Let $X_{1}, \ldots, X_{n}$ be random variables describing some random experiment. We can accumulate these into a random vector $\mathbf{X}=\left(X_{1}, \ldots, X_{n}\right)$. More generally, a collection $\left{X_{t}, t \in \mathscr{T}\right}$ of random variables is called a stochastic process. The set $\mathscr{T}$ is called the parameter set or inder set of the process. It may he discrete (e.g., $\mathbb{N}$ or ${1, \ldots, 10}$ ) or continuous (e.g., $\mathbb{R}{+}=[0, \infty)$ or $\left.[1,10]\right)$. The set of possible values for the stochastic process is called the state space. The joint distribution of $X{1}, \ldots, X_{n}$ is specified by the joint cdf
$$F\left(x_{1}, \ldots, x_{n}\right)=\mathbb{P}\left(X_{1} \leqslant x_{1}, \ldots, X_{n} \leqslant x_{n}\right) .$$
The joint pdf $f$ is given, in the discrete case, by $f\left(x_{1}, \ldots, x_{n}\right)=\mathbb{P}\left(X_{1}=\right.$ $\left.x_{1}, \ldots, X_{n}=x_{n}\right)$, and in the continuous case $f$ is such that
$$\mathbb{P}(\mathbf{X} \in \mathscr{B})-\int_{\mathscr{B}} f\left(x_{1}, \ldots, x_{n}\right) \mathrm{d} x_{1} \ldots \mathrm{d} x_{n}$$
for any (measurable) region $\mathscr{B}$ in $\mathbb{R}^{n}$. The marginal pdfs can be recovered from the joint pdf by integration or summation. For example, in the case of a continuous random vector $(X, Y)$ with joint pdf $f$, the pdf $f_{X}$ of $X$ is found as
$$f_{X}(x)=\int f(x, y) \mathrm{d} y$$
Suppose that $X$ and $Y$ are both discrete or both continuous, with joint pdf $f$, and suppose that $f_{X}(x)>0$. Then the conditional pdf of $Y$ given $X=x$ is given by
$$f_{Y \mid X}(y \mid x)=\frac{f(x, y)}{f_{X}(x)} \quad \text { for all } y .$$
The corresponding conditional expectation is (in the continuous case)
$$\mathbb{E}[Y \mid X=x]=\int y f_{Y \mid X}(y \mid x) \mathrm{d} y$$
Note that $\mathbb{E}[Y \mid X=x]$ is a function of $x$, say $h(x)$. The corresponding random variable $h(X)$ is written as $\mathbb{E}[Y \mid X]$. It can be shown (see, for example, [3]) that its expectation is simply the expectation of $Y$, that is,
$$\mathbb{E}[\mathbb{E}[Y \mid X]]=\mathbb{E}[Y] .$$
When the conditional distribution of $Y$ given $X$ is identical to that of $Y, X$ and $Y$ are said to be independent. More precisely:

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

Consider the experiment where we flip a biased coin $n$ times, with probability $p$ of heads. We can model this experiment in the following way. For $i=1, \ldots, n$, let $X_{i}$ be the result of the $i$-th toss: $\left{X_{i}=1\right}$ means heads (or success), $\left{X_{i}=0\right}$ means tails (or failure). Also, let
$$\mathbb{P}\left(X_{i}=1\right)=p=1-\mathbb{P}\left(X_{i}=0\right), \quad i=1,2, \ldots, n .$$
Last, assume that $X_{1}, \ldots, X_{n}$ are independent. The sequence $\left{X_{i}, i=\right.$ $1,2, \ldots}$ is called a Bernoulli sequence or Bernoulli process with success probability $p$. Let $X=X_{1}+\cdots+X_{n}$ be the total number of successes in $n$ trials (tosses of the coin). Denote by $\mathscr{B}$ the set of all binary vectors $\mathbf{x}=\left(x_{1}, \ldots, x_{n}\right)$ such that $\sum_{i=1}^{n} x_{i}=k$. Note that $\mathscr{B}$ has $\left(\begin{array}{l}n \ k\end{array}\right)$ elements. We now have
\begin{aligned} \mathbb{P}(X=k) &=\sum_{\mathbf{x} \in \mathscr{B}} \mathbb{P}\left(X_{1}=x_{1}, \ldots, X_{n}=x_{n}\right) \ &=\sum_{\mathbf{x} \in \mathscr{B}} \mathbb{P}\left(X_{1}=x_{1}\right) \cdots \mathbb{P}\left(X_{n}=x_{n}\right)=\sum_{\mathbf{x} \in \mathscr{B}} p^{k}(1-p)^{n-k} \ &=\left(\begin{array}{l} n \ k \end{array}\right) p^{k}(1-p)^{n-k} . \end{aligned}
In other words, $X \sim \operatorname{Bin}(n, p)$. Compare this with Example 1.2.
Remark 1.7.1 An infinite sequence $X_{1}, X_{2}, \ldots$ of random variables is called inde pendent if for any finite choice of parameters $i_{1}, i_{2}, \ldots, i_{n}$ (none of them the same the random variables $X_{i_{1}}, \ldots, X_{i_{n}}$ are independent. Many probabilistic models in volve random variables $X_{1}, X_{2}, \ldots$ that are independent and identically distributed abbreviated as iid. We will use this abbreviation throughout this book.

Similar to the one-dimensional case, the expected value of any real-valued function $h$ of $X_{1}, \ldots, X_{n}$ is a weighted average of all values that this function can take Specifically, in the continuous case,
$$\mathbb{E}\left[h\left(X_{1}, \ldots, X_{n}\right)\right]=\int \ldots \int h\left(x_{1}, \ldots, x_{n}\right) f\left(x_{1}, \ldots, x_{n}\right) \mathrm{d} x_{1} \ldots \mathrm{d} x_{n} .$$

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

F(X1,…,Xn)=磷(X1⩽X1,…,Xn⩽Xn).

FX(X)=∫F(X,是)d是

F是∣X(是∣X)=F(X,是)FX(X) 对全部 是.

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