### 统计代写|工程统计作业代写Engineering Statistics代考|Exponential 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|工程统计作业代写Engineering Statistics代考|Continuous Distributions

The exponential (or negative exponential) distribution describes a mechanism whereby the probability of failures (or events) within a time or distance interval depends directly on the number of un-failed items remaining. It describes events such as radioisotope decay, light intensity attenuation through matter of uniform properties, the failure rate of light bulbs, and the residence time or age distribution of particles in a continuous-flow stirred tank. Requirements for the distribution are that, at any time, the probability of any one particular item failing is the same as that of any other item failing and is the same as it was earlier. Another restriction is that the numbers are so large that the measured values seem to be a continuum. The probability distribution functions are
$$p d f(x)=\alpha e^{-\alpha x}, \quad 0 \leq x \leq \infty, \quad \alpha>0$$
and
$$F(x)=1-e^{-\alpha x}$$
The variable $x$ represents the time or distance interval, not the number (or some other measure of quantity) of un-failed items. The argument of an exponential must be dimensionless,

so the units on $\alpha$ are the reciprocal of the units on $x$. This requires that the units on $p d f(x)$ are also the reciprocal of the units on $x$, making $p d f(x)$ be a rate.

Figure $3.8$ illustrates the exponential distribution for $\alpha=0.3$. The mean and variance of the exponential distribution are
$$\mu=\frac{1}{\alpha}$$
and
$$\sigma^{2}=\frac{1}{\alpha^{2}}$$
The continuous random variable $X$ may have any units. The units on $\mu$ will be the same. The units on both $\alpha$ and $f(x)$ are the reciprocal of those of $X . F(x)$ is dimensionless. For a physical interpretation, $\alpha$ represents the fraction of events occurring per unit of space or time. The discrete geometric distribution, in its limit as the number of events is very large and the probability of success is small, approaches the continuous exponential distribution.
Example 3.10: One billion adsorption sites are available on the surface of a solid particle. Gas molecules, randomly and uniformly “looking” for a site, find one upon which to adsorb, which “hides” that site from other molecules. With an infinite gas volume, the rate at which the sites are occupied is therefore proportional to the number of unoccupied sites. If $40 \%$ of the sites are covered within the first 24 hours, how long will it take for $99 \%$ of the adsorption to be complete? What is the average lifetime of an unoccupied site?
From Equation (3.43),
$$40 \%=0.40=F(t)=1-e^{-a t}=1-e^{-a(24)}$$
which gives $\alpha=0.02128440 \ldots$ per hour. From Equation (3.43), $99 \%=0.99=F(t)=1-$ $e^{0.02128 .1}$, which gives $t=216.3636244$… hours or about 9 days. From Equation (3.44), $\mu=$ $1 / \alpha=46.9827645$ hours or almost 2 days.

## 统计代写|工程统计作业代写Engineering Statistics代考|Gamma Distribution

The gamma distribution can represent two mechanisms. In a general situation in which a number of partial random events must occur before a complete event is realized, the probability density function of the complete event is given by the gamma distribution. For instance, rust spots on your car (the partial event), may occur randomly at an average rate of one per month. If 16 spots occur before you decide to have your car repainted (the total event), the gamma distribution is the appropriate one to use to describe the repainting time interval. The gamma distribution is
$$p d f(x)=\frac{\lambda}{\Gamma(\alpha)}(\lambda x)^{\alpha-1} \exp (-\lambda x), x \geq 0$$
and where $\alpha$ and $\lambda>0$ and $\Gamma(\alpha)$ is the gamma function
$$\Gamma(\alpha)=\int_{0}^{\infty} Z^{\alpha-1} e^{-z} d Z$$
The gamma function has several properties
$$\Gamma(\alpha)=(\alpha-1) \Gamma(\alpha-1)$$
and if $\alpha$ is an integer, then
$$\Gamma(\alpha)=(\alpha-1) !$$
The variable $\alpha$ represents the number of partial events required to constitute a complete event, and $\lambda$ is the number of partial events per unit of $x$ (which may be time, distance, space, or item).

If $\alpha=1$, the gamma distribution reduces to the exponential distribution. For that reason, if an event rate is proportional to some power of $x$, then the gamma distribution can also be used as an adjusted exponential distribution. Let’s look at Example $3.10$ again. If adsorption reduces the number of gas molecules available for subsequent adsorption, then the probability of any site being occupied decreases with time. If the frequency with which gas molecules impinge on the particle surface decreases as $(\lambda x)^{a-1}$, then the gamma function describes $f(x)$. However, although close enough for most engineering applications, the power law decrease probably does not describe a real driving force exactly. For such a situation, use of the gamma distribution must be acknowledged as a convenient approximation.
Depending on the values of $\alpha$ and $\lambda, f(x)$ may have various shapes, some of which are illustrated in Figure 3.9. A general analytical expression for $F(x)$ is intractable. For most $\alpha$ values, to obtain the cumulative distribution function, $f(x)$ must be integrated numerically. Excel provides the function GAMMA.DIST $(x, \alpha, 1 / \lambda, 0)$ to return the $p d f(x)$ value

and GAMMA.DIST $(x, \alpha, 1 / \lambda, 1)$ to return the CDF $(x)$ value. Note that the Excel parameter beta is the reciprocal of $\lambda$, here.
The mean and variance of the gamma distribution are
\begin{aligned} \mu &=\frac{\alpha}{\lambda} \ \sigma^{2} &=\frac{\alpha}{\lambda^{2}} \end{aligned}
The units of $X$ are usually count per some interval (time, distance, area, space, or item). Consequently, the units for $\lambda$ are the fraction of total failures per unit of $X$. The coefficient, $\boldsymbol{\alpha}$, is a counting number and is dimensionless, and $f(x)$ has units that are the reciprocal of the units of $X$.

## 统计代写|工程统计作业代写Engineering Statistics代考|Normal Distribution

The normal distribution, often called the Gaussian distribution or bell-shaped error curve, is the most widely used of all continuous probability density functions. The assumption behind this distribution is that any errors (sources of deviation from true) in the experimental results are due to the addition of many independent small perturbation sources. All experimental situations are subject to many random errors and usually yield data that can be adequately described by the normal distribution.

Even if your data is not normally distributed, the averages of data from a nonnormal distribution tend toward being normal. An average of independent samples will have some values above the mean and some below. The average will be close to the mean, and each sample would represent a small independent deviation. In the limit of large sample size, $n$, the standard deviation of the average is related to that of the individual data by $\sigma_{\bar{X}}=\sigma_{X} / \sqrt{n}$. So, when using averages, the normal distribution usually is applicable.
However, this situation is not always true. If you have any doubt that your data are distributed normally, you should use the nonparametric techniques in Chapter 7 to evaluate the distribution. Use of statistics that depend on the normal distribution for a dataset that is distinctly skewed may lead to erroneous results.

An acronym for data that is normally and independently distributed with a mean of $\mu$ and standard deviation of $\sigma$ is $\operatorname{NID}(\mu, \sigma)$.

Regardless of the shape of the distribution of the original population, the central limit theorem allows us to use the normal distribution for descriptive purposes, subject to a single restriction. The theorem simply states that if the population has a mean $\mu$ and a finite variance $\sigma^{2}$, then the distribution of the sample mean $\bar{X}$ approaches the normal distribution with mean $\mu$ and variance $\sigma^{2} / n$ as the sample size $n$ increases. The chief problem with the theorem is how to tell when the sample size is large enough to give reasonable compliance with the theorem. The selection of sample sizes is covered in Chapters 10,11 , and $17 .$
The probability density function $f(x)$ for the normal distribution is
$$f(x)=\frac{1}{\sigma \sqrt{2 \pi}} \mathrm{e}^{\left[-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}\right]},-\infty<x<\infty$$

Note that the argument of the exponentiation, $\left[-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}\right]$, must be dimensionless. As expected, $x, \mu$, and $\sigma$ each have identical units. The exponentiation value is also dimensionless. Also, since $f(x)$ is proportional to $1 / \sigma$, it has the reciprocal units of $x$.

As seen in Equation (3.52), the normal distribution has two parameters, $\mu$ and $\sigma$, which are the mean and standard deviation, respectively. The cumulative distribution function $(C D F)$, described by
$$\operatorname{CDF}(x)=F(x)=P(X \leq x)=\frac{1}{\sigma \sqrt{2 \pi}} \int_{-\infty}^{x} e^{-(X-\mu)^{2} / 2 \sigma^{2}} d X$$
In Equation (3.53) the variable $X$ is the generic variable, and the lower-case $x$ represents a particular value.

The logistic model, $\operatorname{CDF}(x)=F(x)=P(X \leq x)=\frac{1}{1+e^{-s(x-c)}}$, is a convenient and reasonably good approximation to the normal $C D F(x)$. Convenient: It is computationally simple, analytically invertible, and analytically differentiable. Reasonably good: Values are no more different from the normal CDF $(x)$ than that caused by uncertainty on $\mu$ and $\sigma$. For the scale factor, use $s=\sigma / 1.7$, and for the center, use $c=\mu$ (see Exercise 3.15).

## 统计代写|工程统计作业代写Engineering Statistics代考|Continuous Distributions

pdF(X)=一种和−一种X,0≤X≤∞,一种>0

F(X)=1−和−一种X

μ=1一种

σ2=1一种2

40%=0.40=F(吨)=1−和−一种吨=1−和−一种(24)

## 统计代写|工程统计作业代写Engineering Statistics代考|Gamma Distribution

pdF(X)=λΓ(一种)(λX)一种−1经验⁡(−λX),X≥0

Γ(一种)=∫0∞从一种−1和−和d从

Γ(一种)=(一种−1)Γ(一种−1)

Γ(一种)=(一种−1)!

μ=一种λ σ2=一种λ2

## 统计代写|工程统计作业代写Engineering Statistics代考|Normal Distribution

F(X)=1σ2圆周率和[−12(X−μσ)2],−∞<X<∞

CDF⁡(X)=F(X)=磷(X≤X)=1σ2圆周率∫−∞X和−(X−μ)2/2σ2dX

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## MATLAB代写

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