# 数学代写|运筹学作业代写operational research代考|Useful Formulas for the Normal Distribution

## 数学代写|运筹学作业代写operational research代考|Useful Formulas for the Normal Distribution

This appendix lists some useful facts for the standard normal distribution. The formulas for the standard normal density function and the standard normal distribution function are
$$\phi(x)=\frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} x^2} \quad \text { and } \quad \Phi(x)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^x e^{-\frac{1}{2} y^2} d y .$$
The calculation of $\Phi(x)$ seems complicated but does not give any problems in practice. The function can be calculated to every desired degree of accuracy using a rational approximation (a rational function is the quotient of two polynomials). An approximation that is sufficiently accurate for practical purposes is the following:
$$\Phi(x) \approx 1-\frac{1}{2}\left(1+d_1 x+d_2 x^2+d_3 x^3+d_4 x^4+d_5 x^5+d_6 x^6\right)^{-16}, \quad x \geq 0,$$
where
$$\begin{array}{ll} d_1=0.0498673470 & d_4=0.0000380036 \ d_2=0.0211410061 & d_5=0.0000488906 \ d_3=0.0032776263 \quad d_6=0.0000053830 \end{array}$$
The symbol $a \approx b$ means that $a$ is approximately equal to $b$. The absolute error of the approximation is less than $1.5 \times 10^{-7}$. The formula above can be applied directly only for $x \geq 0$. It can also be applied to calculate $\Phi(x)$ for $x<0$ by using the relation $\Phi(x)=1-\Phi(-x)$ for $x<0$.
A common problem is that of solving the equation
$$\Phi(k)=\alpha$$
with $\alpha$ a given number between 0 and 1. The percentile $k$ can be found through a straightforward calculation by using a rational approximation for the inverse function $\Phi^{-1}(\alpha)$. An approximation that is useful in practice is
$$k \approx w-\frac{c_0+c_1 w+c_2 w^2}{1+d_1 w+d_2 w^2+d_3 w^3}, \quad 0.5 \leq \alpha<1$$

## 数学代写|运筹学作业代写operational research代考|The Poisson Process

A stochastic process that is inextricably connected with the Poisson distribution is the Poisson process. This is a counting process that counts the number of occurrences of a particular event over time. The event can be of all kinds: the arrival of customers at a bank, the occurrence of severe earthquakes, the receipt of calls at a telephone exchange, the occurrence of outages at a power plant, the receipt of phone calls at a general emergency number, and so on. In the remainder of this section, we use the terminology of customer arrivals for the occurrence of events over time. When is a counting process a Poisson process? For this, we must assume that the population of potential customers is infinitely large and that customers behave independently of one another.

Definition C.1 (Poisson process). Let the random variable $N(t)$ be the number of customers arriving at a service station up to time $t$. The counting process ${N(t), t \geq 0}$ is called a Poisson process with rate $\lambda, \lambda>0$, if it has the following properties:
(a) $N(0)=0$.
(b) The process has independent increments, that is, the numbers of arrivals in disjoint time intervals are independent of each other.
(c) The number of arrivals in any interval of length $t$ is Poisson distributed with mean $\lambda t$. That is, for all $s, t \geq 0$
$$\mathbb{P}(N(t+s)-N(s)=n)=e^{-\lambda t} \frac{(\lambda t)^n}{n !}, \quad n=0,1, \ldots$$
Property (c) implies that
$\lambda=$ expected value of the number of arrivals
during a time interval of unit length.
The number $\lambda$ is called the arrival rate of the Poisson process.
The Poisson process provides a good model description in many real-world situations. The explanation for this lies in the following, roughly formulated, result. Suppose that at the micro level, there are numerous stochastic processes that are independent of one another and that each generate arrivals sparsely. Then one can show that at the macro level, the resultant of all these processes is approximately a Poisson process. This explains, for example, why the arrival process of customers at a post office can often be described by a Poisson process. The population of potential customers is very large: every individual behaves according to a specific pattern, but the combination of all these typically independent patterns leads to an unpredictable whole that can be described by a Poisson process.

# 运筹学代考

## 数学代写|运筹学作业代写operational research代考|Useful Formulas for the Normal Distribution

$$\phi(x)=\frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} x^2} \quad \text { and } \quad \Phi(x)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^x e^{-\frac{1}{2} y^2} d y$$

$$\Phi(x) \approx 1-\frac{1}{2}\left(1+d_1 x+d_2 x^2+d_3 x^3+d_4 x^4+d_5 x^5+d_6 x^6\right)^{-16}, \quad x \geq 0$$

$$d_1=0.0498673470 \quad d_4=0.0000380036 d_2=0.0211410061 \quad d_5=0.0000488906 d_3$$

$$\Phi(k)=\alpha$$

$$k \approx w-\frac{c_0+c_1 w+c_2 w^2}{1+d_1 w+d_2 w^2+d_3 w^3}, \quad 0.5 \leq \alpha<1$$

## 数学代写|运筹学作业代写operational research代考|The Poisson Process

(a) $N(0)=0$.
(b) 过程具有独立增量，即在不相交的时间间隔内到达的次数相互独立。
(c) 任何长度间隔内的到达次数 $t$ 服从均值的泊松分布 $\lambda t$. 也就是说，对于所有 $s, t \geq 0$
$$\mathbb{P}(N(t+s)-N(s)=n)=e^{-\lambda t} \frac{(\lambda t)^n}{n !}, \quad n=0,1, \ldots$$

$\lambda=$

myassignments-help数学代考价格说明

1、客户需提供物理代考的网址，相关账户，以及课程名称，Textbook等相关资料~客服会根据作业数量和持续时间给您定价~使收费透明，让您清楚的知道您的钱花在什么地方。

2、数学代写一般每篇报价约为600—1000rmb，费用根据持续时间、周作业量、成绩要求有所浮动(持续时间越长约便宜、周作业量越多约贵、成绩要求越高越贵)，报价后价格觉得合适，可以先付一周的款，我们帮你试做，满意后再继续，遇到Fail全额退款。

3、myassignments-help公司所有MATH作业代写服务支持付半款，全款，周付款，周付款一方面方便大家查阅自己的分数，一方面也方便大家资金周转，注意:每周固定周一时先预付下周的定金，不付定金不予继续做。物理代写一次性付清打9.5折。

Math作业代写、数学代写常见问题

myassignments-help擅长领域包含但不是全部:

myassignments-help服务请添加我们官网的客服或者微信/QQ，我们的服务覆盖：Assignment代写、Business商科代写、CS代考、Economics经济学代写、Essay代写、Finance金融代写、Math数学代写、report代写、R语言代考、Statistics统计学代写、物理代考、作业代写、加拿大代考、加拿大统计代写、北美代写、北美作业代写、北美统计代考、商科Essay代写、商科代考、数学代考、数学代写、数学作业代写、physics作业代写、物理代写、数据分析代写、新西兰代写、澳洲Essay代写、澳洲代写、澳洲作业代写、澳洲统计代写、澳洲金融代写、留学生课业指导、经济代写、统计代写、统计作业代写、美国Essay代写、美国代考、美国数学代写、美国统计代写、英国Essay代写、英国代考、英国作业代写、英国数学代写、英国统计代写、英国金融代写、论文代写、金融代考、金融作业代写。

Scroll to Top