### 统计代写|工程统计代写engineering statistics代考|Rectangular Distribution

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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|ALTERNATE REPRESENTATIONS

The range of the distribution is symmetric around the origin (say $-a$ to $+a$ ), or centered around a fixed constant (say $\theta-1 / 2, \theta+1 / 2$ ) in several practical applications:
$$f(x ; a, \theta)=\left{\begin{array}{rll} 1 / \theta & \text { for } & a \leq x \leq a+\theta \ 1 /(2 \theta) & \text { for } & a-\theta \leq x \leq a+\theta \ 0 & & \text { otherwise. } \end{array}\right.$$
For $a=1$ we get $f(x ; a, b)=1 / 2$ for $-1<x<1$; and for $a=1 / 2$ we get $f(u ; a, b)=1$ for $-1 / 2<u<1 / 2$

The mean and variance are $\mu=(a+b) / 2$, and $\sigma^{2}=(b-a)^{2} / 12$, as shown on page 21 . Write $\mu=(a+b) / 2$ and $\sigma=(b-a) /(2 \sqrt{3})$. Cross multiply to get $(a+b)=2 \mu$, and $(b-$ $a)=(2 \sqrt{3}) \sigma$. Add them to get $b=\mu+\sqrt{3} \sigma$. Subtracting gives $a=\mu-\sqrt{3} \sigma$, from which $(b-a)=(2 \sqrt{3}) \sigma$. Thus, the PDF becomes
$$f(x ; \mu, \sigma)=1 /(2 \sqrt{3} \sigma), \mu-\sqrt{3} \sigma \leq x \leq \mu+\sqrt{3} \sigma$$
The CDF is
F(x ; a, b)=\left{\begin{aligned} 0 & \text { for } x<\mu-\sqrt{3} \sigma \ 0.50(1+(x-\mu) /(\sqrt{3} \sigma)) & \text { for } \mu-\sqrt{3} \sigma \leq x \leq \mu+\sqrt{3} \sigma \ 1 & \text { for } x>\mu+\sqrt{3} \sigma \end{aligned}\right.
and inverse CDF is $F^{-1}(p)=\mu+\sqrt{3} \sigma(2 p-1)$ for $0<p<1$. Put $\sigma=1$ to get the standardized $\operatorname{CUNI}(-\sqrt{3},+\sqrt{3})$ that has mean zero and variance unity.

Some applications in engineering, theoretical computer science, and number theory use the uniform distribution modulo $k$. This allows the distribution to be extended to the entire real line (because the “mod $\mathrm{k}$ ” maps all such real numbers to $(0, k)$ range), and are more applicable to discrete uniform distribution. The uniform distribution on a circle has PDF $f(x)=1 /(2 \pi)$, for $0<x \leq 2 \pi$.

## 统计代写|工程统计代写engineering statistics代考|RELATED DISTRIBUTIONS

Due to its relationship with many other distributions, it is extensively used in computer generation of random variables. $U(0,1)$ is a special case of Beta-I $(a, b)$ when $a=b=1$. If $X \sim U(0,1)$ then $Y=-\log (X) \sim \mathrm{SED}$ (i.e., $\operatorname{EXP}(1))$, and $Y=-2 \log (X)$ has a $\chi_{2}^{2}$ distribution. If $x_{1}, x_{2}, \ldots, x_{k}$ are independent samples from possibly $k$ different $\mathrm{U}(0,1)$ populations,

$P_{k}=\sum_{j=1}^{k}-2 \ln \left(x_{j}\right)$ being the sum of $k$ IID $\chi_{2}^{2}$ variates has $\chi_{2 k}^{2}$ distribution. This is called Pearson’s statistic in tests of significance [114]. A simple change of variable transformation $Y=(X-a) /(b-a)$ in the general PDF results in the SUD (i.e., $\mathrm{U}(0,1)) . \mathrm{U}(0,1)$ is also related to arcsine distribution as $Y=-\cos (\pi U / 2)$ (Chapter 5). If $X$ is any continuous random variable with CDF $F(x)$, then $U=F(x) \sim U[0,1]$.
Example 2.1 Distribution of $F(x)$
If $X$ is a continuous variate, find the distribution of $U=F(x)$.
Solution 2.2 Consider
$$\mathrm{F}(\mathrm{u})=\operatorname{Pr}(\mathrm{U} \leq u)=\operatorname{Pr}(\mathrm{F}(\mathrm{x}) \leq u)=\operatorname{Pr}\left(\mathrm{x} \leq \mathrm{F}^{-1}(u)\right)=\mathrm{F}\left[\mathrm{F}^{-1}(u)\right]=\mathrm{u}$$
The CDF of a rectangular distribution $\operatorname{CUNI}(a, b)$ is $(x-a) /(b-a)$. Put $a=0, b=1$ to get $F(x)=x$. Equation (2.6) then shows that $U$ is an SUD.

This property can be used to generate random numbers from a distribution if the expression for its CDF (or SF) involves simple or invertible arithmetic or transcendental functions. For example, the CDF of an exponential distribution is $F(x)=1-\exp (-\lambda x)$. Equating to a random number $u$ in the range $[0,1]$ and solving for $x$, we get $1-e^{-\lambda x}=u$ or $x=-\log (1-u) / \lambda$, using which random numbers from exponential distributions can be generated.

Problem 2.3 If a doctor and a nurse arrive a hospital independently and uniformly during $8 \mathrm{AM}$ and $9 \mathrm{AM}$, find the probability that the first patient to arrive has to wait longer than 10 $\mathrm{min}$, if consultation is possible only when both the doctor and nurse are in office.

Problem 2.4 If three random variables $X, Y, Z$ are independently and uniformly distributed over $(0,1)$, show that $\operatorname{Pr}(X>Y Z)=3 / 4$

## 统计代写|工程统计代写engineering statistics代考|PROPERTIES OF RECTANGULAR DISTRIBUTION

This distribution has a special type of symmetry called flat-symmetry. Hence, all odd central moments $\mu_{2 r+1}$ except the first one are zeros. The median always coincides with the mean, and the mode can be any value within the range. As the probability is constant throughout the interval, the range is always finite (and quite often small). As $F(x)=(x-a) /(b-a)$, its inverse is
$$F^{-1}(p)=a+p(b-a), \quad \text { for } 0<p<1 .$$
A uniform distribution defined in an interval $(c, c+\theta)$ has PDF
$$f(x ; \theta)=1 / \theta \text { for } c \leq x \leq c+\theta .$$

Take $c=0$ to get the standard form $f(x ; \theta)=1 / \theta, 0<x<\theta$. This is the analogue of the $\operatorname{DUNI}(\mathrm{N})$ with probability function $f(x ; N)=1 / N$, for $x=0,1,2, \ldots, N-1$ discussed in Chapter 3 of Chattamvelli and Shanmugam (2020) [42]. The transformation $Y=(b-a)-X$ results in the same distribution. In particular, if $X \sim U(0,1)$ then $Y=1-X \sim U(0,1)$. This property of $U(0,1)$ is used in generating random samples from other distributions like the exponential distribution (page 35). Only the extremes of a sample $x_{(1)}$ and $x_{(n)}$ are sufficient to fit this distribution.
Problem 2.18 If $X \sim \mathrm{U}[0,1]$, find the distribution of $Y=\exp (X)$, and its variance.
Problem $2.19$ If $X \sim U(0,1)$, find the distribution of $Y=X /(1-X)$.

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

$$f(x ; a, \theta)=\left{ 1/θ 为了 一个≤X≤一个+θ 1/(2θ) 为了 一个−θ≤X≤一个+θ 0 否则。 \正确的。$$

F(X;μ,σ)=1/(23σ),μ−3σ≤X≤μ+3σ
CDF 是
$$F(x ; a, b)=\left{ 0 为了 X<μ−3σ 0.50(1+(X−μ)/(3σ)) 为了 μ−3σ≤X≤μ+3σ 1 为了 X>μ+3σ\正确的。$$

## 统计代写|工程统计代写engineering statistics代考|RELATED DISTRIBUTIONS

F(在)=公关⁡(在≤在)=公关⁡(F(X)≤在)=公关⁡(X≤F−1(在))=F[F−1(在)]=在

## 统计代写|工程统计代写engineering statistics代考|PROPERTIES OF RECTANGULAR DISTRIBUTION

F−1(p)=一个+p(b−一个), 为了 0<p<1.

F(X;θ)=1/θ 为了 C≤X≤C+θ.

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