### 统计代写|工程统计代写engineering statistics代考|STAT2110

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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|工程统计代写engineering statistics代考|INTRODUCTION

The beta distribution has a long history that can be traced back to the year 1676 in a letter from Issac Newton to Henry Oldenbeg (see Dutka (1981) [55]). It is widely used in civil, geotechnical, earthquake, and metallurgical engineering due to its close relationship with other continuous distributions. The PDF of Beta-I $(a, b)$ is given by ${ }^{1}$
$$f(x ; a, b)=x^{a-1}(1-x)^{b-1} / B(a, b)$$
where $00$ and $b>0$ results in a variety of distributional shapes. The Beta-I distribution is a proper choice in risk-modeling because the risks in many applications can be lower and upper bounded, and scaled to any desired range (say $(0,1)$ range) [78]. Events constrained to happen within a finite interval can be modeled due to the wide variety of shapes assumed by this distribution.

It is also used in Bayesian models with unknown probabilities, in order-statistics and reliability analysis. In Bayesian analysis, the prior distribution is assumed to be Beta-I for binomial proportions. It is used to model the proportion of fat (by weight) in processed or canned food, percentage of impurities in some manufactured products like food items, cosmetics, laboratory chemicals, etc. Data in the form of proportions arise in many applied fields like marketing, toxicology, bioinformatics, genomics, etc. Beta distribution is the preferred choice when these quantities exhibit extra variation than expected. Important distributions belonging to the beta family are discussed below. These include type I and type-II beta distributions. We will use the respective notations $\operatorname{Beta}-\mathrm{I}(a, b)$, and $\operatorname{Beta}-\mathrm{II}(a, b) .{ }^{2}$ Beta distributions with three or more parameters are also briefly mentioned.

## 统计代写|工程统计代写engineering statistics代考|ALTERNATE REPRESENTATIONS

Write $c=a-1$ and $d=b-1$ to get the alternate form
$$f(x ; c, d)=x^{c}(1-x)^{d} / B(c+1, d+1) .$$ Put $x=\sin ^{2}(\theta)$ in (4.2) to get
$$f(\theta ; c, d)=\sin ^{2 c}(\theta) \cos ^{2 d}(\theta) / B(c+1, d+1) \text { for } 0<\theta<\pi / 2$$
Some applications use $a$ and $n-a+1$ as parameters resulting in
$$f(x ; a, b, n)=x^{a-1}(1-x)^{n-a} / B(a, n-a+1) .$$
A symmetric beta distribution results when $a=b$ with PDF
$$f(x ; a)=x^{a-1}(1-x)^{a-1} / B(a, a)=[x(1-x)]^{a-1} \Gamma(2 a) /[\Gamma(a)]^{2} .$$
Beta distributions defined on $(-1,+1)$ are encountered in some applications. Using the transformation $Y=2 X-1$ we get $f(y)=f(x) / 2=f((y+1) / 2) / 2$. This results in the PDF
$$f(y ; a, b)=[(y+1) / 2]^{a-1}[(1-y) / 2]^{b-1} /[2 B(a, b)] .$$
This simplifies to
$\left.f(y ; a, b)=C(1+y)^{a-1}(1-y)\right]^{b-1}$ where $-1<y<1$, and $C=1 /\left[2^{a+b-1} B(a, b)\right] .$
This also can be generalized to 4-parameters as
$$\left.f(x ; a, b, c, d)=C(1+x / c)^{a-1}(1-x / d)\right]^{b-1}$$
and to the 6-parameters as
$$\left.f(x ; a, b, c, d, p, q)=C(1+(x-p) / c)^{a-1}(1-(x-q) / d)\right]^{b-1},$$
where $C$ is the normalizing constant, which is found using the well-known integral
$$\int_{a}^{b}(x-a)^{a-1}(b-x)^{b-1} d x=(b-a)^{a+b-1} B(a, b) .$$
These are related to the Berstein-type basis functions $Y_{k}^{n}(x ; a, b, m)=\left(\begin{array}{c}m \ k\end{array}\right)(x-a)^{k}(b-$ $x)^{n-k} /(b-a)^{m}[147]$. Truncated and size-biased versions of them are used in several engineering fields.

## 统计代写|工程统计代写engineering statistics代考|RELATION TO OTHER DISTRIBUTIONS

It is a special case of gamma distribution with $m=1$. It reduces to uniform (rectangular) distribution $\mathrm{U}(0,1)$ for $a=b=1$. A triangular-shaped distribution results for $a=1$ and $b=2$, or vice versa. When $a=b=1 / 2$, this distribution reduces to the arcsine distribution of first kind (Chapter 5). If $b=1$ and $a \neq 1$, it reduces to power-series distribution $f(x ; a)=a x^{a-1}$ using the result $\Gamma(a+1)=a * \Gamma(a)$. A J-shaped distribution is obtained when $a$ or $b$ is less than one. If $X$ and $Y$ are IID gamma distributed with parameters $a$ and $b$, the ratio $Z=X /(X+Y)$ is Beta-I $(a, b)$ distributed. If $X_{k}^{\prime}$ s are IID Beta-I $\left(\frac{2 k-1}{2 n}, \frac{1}{2 n}\right)$ random variables, the distribution of the geometric mean (GM) of them $Y=\left(\prod_{k=1}^{n} X_{k}\right)^{17 n}$ is SASD-I distributed (Chapter 5) for $n \geq 2([90],[22])$. This has the interpretation that the GM of Beta-I $\left(\frac{2 k-1}{2 n}, \frac{1}{2 n}\right)$ random variables converges to arcsine law whereas the AM tends to the normal law (central limit theorem). As $\chi^{2}$ distribution is a special case of gamma distribution, a similar result follows as $Z=\chi_{m}^{2} /\left(\chi_{m}^{2}+\chi_{n}^{2}\right) \sim \operatorname{Beta}-\mathrm{I}(m / 2, n / 2)$

As $\left(\chi_{m}^{2}+\chi_{n}^{2}\right)$ is independent of $Z$, the above result can be generalized as follows: If $X_{1}, X_{2}, \ldots X_{n}$ are IID normal variates with zero means and variance $\sigma_{k}^{2}$, then $Z_{1}=X_{1}^{2} /\left(X_{1}^{2}+\right.$ $\left.X_{2}^{2}\right), Z_{2}=\left(X_{1}^{2}+X_{2}^{2}\right) /\left(X_{1}^{2}+X_{2}^{2}+X_{3}^{2}\right)$, and so on are mutually independent beta random variables. If $X$ has an $F(m, n)$ distribution, then $Y=(m / n) X /[1+(m / n) X]$ is beta distributed. The beta distribution is also related to the Student’s $t$ distribution under the transformation $x=1 /\left(1+t^{2} / n\right)$. Similarly, $y=-\log (x)$ has PDF
$$f(y ; a, b)=\exp (-a y)(1-\exp (-y))^{b-1}$$
and $y=x /(1-x)$ results in beta-prime distribution (page 48). The positive eigenvalue of Roy’s $\theta_{\max }$-criterion used in MANOVA has a Beta-I distribution when $s=\max \left(p, n_{h}\right)=1$, where $p$ is the dimensionality and $n_{h}$ is the DoF of the hypothesis.

Problem 4.1 Prove that $B(a+1, b)=[a /(a+b)] B(a, b)$ where $B(a, b)$ denotes the CBF. What is the value of $B(.5, .5)$ ?

Problem 4.2 If $X \sim \operatorname{Beta}-\mathrm{I}(a, b)$, find the distribution of $Y=(1-X) / X$, and obtain its mean and variance. Find the ordinary moments.

Problem 4.3 If $X$ and $Y$ are independent gamma random variables GAMMA $(a, \lambda)$ and $\operatorname{GAMMA}(b, \lambda)$, then prove that $X /(X+Y)$ is $\operatorname{Beta}(a, b)$

Problem 4.4 Verify whether $f(x ; c, d)=(1+x)^{c-1}(1-x)^{d-1} /\left[2^{c+d-1} \mathrm{~B}(c, d)\right]$ is a PDF for $-1<x<1$, where $B(c, d)$ is the complete beta function.

## 统计代写|工程统计代写engineering statistics代考|INTRODUCTION

Beta 分布有很长的历史，可以追溯到 1676 年 Issac Newton 给 Henry Oldenbeg 的一封信（参见 Dutka (1981) [55]）。由于与其他连续分布关系密切，它被广泛应用于土木、岩土、地震和冶金工程。Beta-I 的 PDF(一个,b)是（谁）给的1

F(X;一个,b)=X一个−1(1−X)b−1/乙(一个,b)

## 统计代写|工程统计代写engineering statistics代考|ALTERNATE REPRESENTATIONS

F(X;C,d)=XC(1−X)d/乙(C+1,d+1).放X=罪2⁡(θ)在 (4.2) 中得到

F(θ;C,d)=罪2C⁡(θ)因2d⁡(θ)/乙(C+1,d+1) 为了 0<θ<圆周率/2

F(X;一个,b,n)=X一个−1(1−X)n−一个/乙(一个,n−一个+1).

F(X;一个)=X一个−1(1−X)一个−1/乙(一个,一个)=[X(1−X)]一个−1Γ(2一个)/[Γ(一个)]2.
Beta 分布定义于(−1,+1)在某些应用程序中遇到。使用转换是=2X−1我们得到F(是)=F(X)/2=F((是+1)/2)/2. 这导致PDF

F(是;一个,b)=[(是+1)/2]一个−1[(1−是)/2]b−1/[2乙(一个,b)].

F(是;一个,b)=C(1+是)一个−1(1−是)]b−1在哪里−1<是<1， 和C=1/[2一个+b−1乙(一个,b)].

F(X;一个,b,C,d)=C(1+X/C)一个−1(1−X/d)]b−1

F(X;一个,b,C,d,p,q)=C(1+(X−p)/C)一个−1(1−(X−q)/d)]b−1,

∫一个b(X−一个)一个−1(b−X)b−1dX=(b−一个)一个+b−1乙(一个,b).

## 统计代写|工程统计代写engineering statistics代考|RELATION TO OTHER DISTRIBUTIONS

F(是;一个,b)=经验⁡(−一个是)(1−经验⁡(−是))b−1

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## MATLAB代写

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