### 统计代写|工程统计代写engineering statistics代考|ENGG 202

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• 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|TYPE-II BETA DISTRIBUTION

Beta distribution of the second kind (also called type-II beta distribution, beta-prime distribution, or inverted beta distribution (IBD)) is obtained from the above by the transformation $Y=X /(1-X)$ or equivalently $X=Y /(1+Y)$. When $x \rightarrow 0, y \rightarrow 0$, and when $x \rightarrow 1$, $y \rightarrow \infty$. Hence, the range of $Y$ is from 0 to $\infty$. The PDF is given by
$$f(y ; a, b)=y^{a-1} /\left[B(a, b)(1+y)^{a+b}\right], \quad y>0, a, b>0 .$$
The Beta-I distribution is used to model random experiments or occurrences that vary between two finite limits, that are mapped to the $(0,1)$ range, while Beta-II is used when upper limit is infinite. It is also used in risk analysis in finance and marketing, etc.

Put $a=b=1$ to get Beta( $(1,1)$, which is identical to $\mathrm{U}(0,1)$ distribution. If $X$ is Beta$\mathrm{I}(a, b)$ then $(1-X) / X$ is $\operatorname{Beta}-\mathrm{II}(b, a)$, and $X /(1-X)$ is $\operatorname{Beta-II}(a, b)$. If $X$ and $Y$ are independent gamma random variables GAMMA $(a, \lambda)$ and GAMMA $(b, \lambda)$, then $X /(X+Y)$ is $\operatorname{Beta}(a, b)$. As gamma and $\chi^{2}$ are related, this result can also be stated in terms of normal variates as follows. If $X$ and $Y$ are independent normal variates, then $Z=X^{2} /\left(X^{2}+Y^{2}\right)$ is Beta-I distributed. In addition, if $X_{1}, X_{2}, \ldots, X_{k}$ are IID $N(0,1)$ and $Z_{1}=X_{1}^{2} /\left(X_{1}^{2}+X_{2}^{2}\right)$, $Z_{2}=\left(X_{1}^{2}+X_{2}^{2}\right) /\left(X_{1}^{2}+X_{2}^{2}+X_{3}^{2}\right)$, and so on, $Z_{j}=\sum_{i=1}^{j} X_{i}^{2} / \sum_{i=1}^{j+1} X_{i}^{2}$, then each of them are Beta-I distributed, as also the product of any consecutive set of $Z_{j}^{\prime}$ s are beta distributed. The logistic distribution and type II beta distribution are related as $Y=-\ln (\mathrm{X})$. If $X$ is $\mathrm{Beta}-\mathrm{I}(a, b)$ then $Y=\ln (X /(1-X))$ has a generalized logistic distribution. Dirichlet distribution is a gencralization of beta distribution. Order statistic from uniform distribution is beta distributed. In general, $j^{t h}$ highest order statistic from a uniform distribution is $\operatorname{Beta}-\mathrm{I}(j, n-j+1)$.

## 统计代写|工程统计代写engineering statistics代考|MOMENTS AND GENERATING FUNCTIONS OF TYPE-II BETA

The mean and variance are $\mu=a /(b-1)$ and $\sigma^{2}=a(a+b-1) /\left[(b-1)^{2}(b-2)\right]$ for $b>2$. Consider $\mathrm{E}\left(Y^{k}\right)$
$$\int_{0}^{\infty} y^{k} f_{y}(a, b) d y=\int_{0}^{\infty} y^{a+k-1} /\left[B(a, b)(1+y)^{a+b}\right] d y$$
Put $x=y /(1+y)$ so that $y=x /(1-x),(1+y)=1 /(1-x)$ and $d y / d x=[(1-x)-$ $x(-1)] /(1-x)^{2}$. This simplifies to $1 /(1-x)^{2}$. The range of $X$ is $[0,1]$. Hence, $(4.22)$ becomes
$$(1 / B(a, b)) \int_{0}^{\infty} y^{a+k-1} /(1+y)^{a+b} d y=(1 / B(a, b)) \int_{0}^{1} x^{a+k-1}(1-x)^{b-k-1} d x .$$
This is $B(a+k, b-k) / B(a, b)$. Put $k=1$ to get the mean as $\Gamma(a+1) \Gamma(b-1) \Gamma(a+$ b) $/[\Gamma(a) \Gamma(b) \Gamma(a+b)]$. Write $\Gamma(a+1)=a \Gamma(a)$ in the numerator, and $\Gamma(b)=(b-1) \Gamma(b-$ 1) in the denominator and cancel out common factors to get $\mu=a /(b-1)$. Put $k=2$ to get the second moment as $B(a+2, b-2) / B(a, b)=\Gamma(a+2) \Gamma(b-2) \Gamma(a+b) /[\Gamma(a) \Gamma(b) \Gamma(a+$ $b)]=a(a+1) /[(b-1)(b-2)]$. From this the variance is obtained as $a(a+1) /[(b-1)(b-$ 2)] $-a^{2} /(b-1)^{2}$. Take $\mu=a /(b-1)$ as a common factor. This can now be written as $\mu\left(\frac{a+1}{b-2}-\mu\right)$. Substitute for $\mu$ inside the bracket and take $(b-1)(b-2)$ as common denominator. The numerator simplifies to $b-a+2 a-1=(a+b-1)$. Hence, the variance becomes $\sigma^{2}=a(a+b-1) /\left[(b-1)^{2}(b-2)\right]$. As $(a+1) /(b-2)-\mu=(a+b) /[(b-1)(b-2)]$, this expression is valid for $b>2$. Unlike the Beta-I distribution whose variance is always bounded, the variance of Beta-II can be increased arbitrarily by keeping b constant (say near $\left.2^{+}\right)$and letting $a \rightarrow \infty$. It can also be decreased arbitrarily when $(a+1) /(b-2)$ tends to $\mu=a /(b-1)$. The expectation of $[X /(1-X)]^{k}$ is easy to compute in terms of complete gamma function as $\mathrm{E}[X /(1-X)]^{k}=\frac{\Gamma(a+k) \Gamma(b-k)}{\Gamma(a) \Gamma(b)}$. See Table $4.2$ for further properties.

Example 4.14 The mode of Beta-II distribution Prove that the mode of Beta-II distribution is $(a-1) /(b+1)$.
Solution 4.15 Differentiate the PDF (without constant multiplier) w.r.t. $y$ to get
$$f^{\prime}(y)=\left[(1+y)^{a+b}(a-1) y^{a-2}-y^{a-1}(a+b)(1+y)^{a+b-1}\right] /(1+y)^{2(a+b)}$$

Equate the numerator to zero and solve for $y$ to get $y[a+b-a+1]=(a-1)$, or $y=(a-$ 1) $/(b+1)$As the Beta-I random variable takes values in $[0,1]$, any CDF can be substituted for $x$ to get a variety of new distributions (Chattamvelli (2012) [36]). For instance, put $x=\Phi(x)$, the CDF of a normal variate to get the beta-normal distribution with PDF
$$f(x ; a, b)=(1 / B[a, b]) \phi(x)[\Phi(x)]^{a-1}[1-\Phi(x)]^{b-1}$$

## 统计代写|工程统计代写engineering statistics代考|TAIL AREAS USING IBF

Tail areas of several statistical distributions are related to the beta CDF, as discussed below. The survival function of a binomial distribution $\operatorname{BINO}(n, p)$ is related to the left tail areas of Beta-I distribution as:
$$\sum_{x=a}^{n}\left(\begin{array}{l} n \ x \end{array}\right) p^{x} q^{n-x}=\mathrm{I}{p}(a, n-a+1)$$ Using the symmetry relationship, the CDF becomes $$\sum{x=0}^{a-1}\left(\begin{array}{l} n \ x \end{array}\right) p^{x} q^{n-x}=\mathrm{I}_{q}(n-a+1, a) .$$

When both $a$ and $b$ are integers, this has a compact representation as
$$\mathrm{I}{x}(a, b)=1-\sum{k=0}^{a-1}\left(\begin{array}{c} a+b-1 \ k \end{array}\right) x^{k}(1-x)^{a+b-1-k} .$$
The survival function of negative binomial distribution is related as follows:
$$\sum_{x=a}^{n}\left(\begin{array}{c} n+x-1 \ x \end{array}\right) p^{n} q^{x}=\mathrm{I}{q}(a, n)=1-\mathrm{I}{p}(n, a)$$
The relationship between the CDF of central $F$ distribution and the IBF is
$$\mathrm{F}{m, n}(x)=\mathrm{I}{y}(m / 2, n / 2),$$
where $(m, n)$ are the numerator and denominator $\mathrm{DoF}$ and $y=m x /(n+m x)$. Similarly, Student’s $t$ CDF is evaluated as
$$\mathrm{T}{n}(t)=(1 / 2)\left(1+\operatorname{sign}(\mathrm{t}) \mathrm{I}{x}(1 / 2, n / 2)\right)=(1 / 2)\left(1+\operatorname{sign}(\mathrm{t})\left[1-\mathrm{I}_{y}(n / 2,1 / 2)\right]\right),$$
where $x=t^{2} /\left(n+t^{2}\right), y=1-x=n /\left(n+t^{2}\right), \operatorname{sign}(\mathrm{t})=+1$ if $\mathrm{t}>0,-1$ if $\mathrm{t}<0$ and is zero for $t=0$.

The IBF is related to the tail areas of binomial, negative binomial, Student’s $t$, central $F$ distributions. It is also related to the confluent hypergeometric function, generalized logistic distribution, the distribution of order statistics from uniform populations, and the Hotelling’s $\mathrm{T}^{2}$ statistic. The hypergeometric function can be approximated using the IBF also [145]. The Dirichlet (and its inverse) distribution can be expressed in terms of IBF [140]. It is related to the CDF of noncentral distributions. For instance, the CDF of singly noncentral beta (Seber (1963) [121]), singly type-II noncentral beta, and doubly noncentral beta (Chattamvelli (1995) [31]), noncentral T (Chattamvelli (2012) [36], Craig (1941) [48]), noncentral F (Chattamvelli (1996) [33], Patnaik (1949) [107]), and the sample multiple correlation coefficient (Ding and Bargmann (1991) [53], Ding (1996) [52]) could all be evaluated as infinite mixtures of IBF. It is used in string theory to calculate and reproduce the scattering amplitude in terms of Regge trajectories, and to model properties of strong nuclear force.

## 统计代写|工程统计代写engineering statistics代考|TYPE-II BETA DISTRIBUTION

F(是;一个,b)=是一个−1/[乙(一个,b)(1+是)一个+b],是>0,一个,b>0.
Beta-I 分布用于模拟在两个有限限制之间变化的随机实验或事件，这些限制映射到(0,1)范围，而 Beta-II 用于上限为无限时。它还用于金融和营销等领域的风险分析。

## 统计代写|工程统计代写engineering statistics代考|MOMENTS AND GENERATING FUNCTIONS OF TYPE-II BETA

∫0∞是ķF是(一个,b)d是=∫0∞是一个+ķ−1/[乙(一个,b)(1+是)一个+b]d是

(1/乙(一个,b))∫0∞是一个+ķ−1/(1+是)一个+bd是=(1/乙(一个,b))∫01X一个+ķ−1(1−X)b−ķ−1dX.

F′(是)=[(1+是)一个+b(一个−1)是一个−2−是一个−1(一个+b)(1+是)一个+b−1]/(1+是)2(一个+b)

F(X;一个,b)=(1/乙[一个,b])φ(X)[披(X)]一个−1[1−披(X)]b−1

## 统计代写|工程统计代写engineering statistics代考|TAIL AREAS USING IBF

∑X=一个n(n X)pXqn−X=我p(一个,n−一个+1)使用对称关系，CDF 变为

∑X=0一个−1(n X)pXqn−X=我q(n−一个+1,一个).

∑X=一个n(n+X−1 X)pnqX=我q(一个,n)=1−我p(n,一个)

F米,n(X)=我是(米/2,n/2),

IBF 与二项式、负二项式、Student’s 的尾部区域有关吨, 中央F分布。它还与汇合的超几何函数、广义逻辑分布、均匀总体的顺序统计分布以及 Hotelling 的吨2统计。超几何函数也可以使用 IBF 来近似 [145]。Dirichlet（及其逆）分布可以用 IBF [140] 来表示。它与非中心分布的 CDF 有关。例如，单非中心 beta (Seber (1963) [121])、单 II 型非中心 beta 和双重非中心 beta (Chattamveli (1995) [31])、非中心 T (Chattamveli (2012) [36] 的 CDF , Craig (1941) [48]), noncentral F (Chattamveli (1996) [33], Patnaik (1949) [107]) 和样本多重相关系数 (Ding and Bargmann (1991) [53], Ding (1996) ) [52]) 都可以被评估为 IBF 的无限混合。它在弦理论中用于根据雷格轨迹计算和再现散射幅度，并模拟强核力的特性。

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## MATLAB代写

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