### 统计代写|生物统计代写biostatistics代考| PROBABILITY MODELS

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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|生物统计代写biostatistics代考|The Binomial Probability Model

The binomial probability model can be used for modeling the number of times a particular event occurs in a sequence of repeated trials. In particular, a binomial random variable is a discrete variable that is used to model chance experiments involving repeated dichotomous trials. That is, the binomial model is used to model repeated trials where the outcome of each trial is one of the two possible outcomes. The conditions under which the binomial probability model can be used are given below.

A random variable satisfying the above conditions is called a binomial random variable. Note that a binomial random variable $X$ simply counts the number of successes that occurred in $n$ trials. The probability distribution for a binomial random variable $X$ is given by the mathematical expression
$$p(x)=\frac{n !}{x !(n-x) !} p^{x}(1-p)^{n-x} \quad \text { for } x=0,1, \ldots, n$$
where $p(x)$ is the probability that $X$ is equal to the value $x$. In this formula

• $\frac{n !}{x !(n-x) !}$ is the number of ways for there to be $x$ successes in $n$ trials,
• $n !=n(n-1)(n-2) \cdots 3 \cdot 2 \cdot 1$ and $0 !=1$ by definition,
• $p$ is the probability of a success on any of the $n$ trials,
• $p^{x}$ is the probability of having $x$ successes in $n$ trials,
• $1-p$ is the probability of a failure on any of the $n$ trials,
• $(1-p)^{n-x}$ is the probability of getting $n-x$ failures in $n$ trials.
Examples of the binomial distribution are given in Figure 2.24. Note that a binomial distribution will have a longer tail to the right when $p<0.5$, a longer tail to the left when $p>0.5$, and is symmetric when $p=0.5$.

Because the computations for the probabilities associated with a binomial random variable are tedious, it is best to use a statistical computing package such as MINITAB for computing binomial probabilities.

## 统计代写|生物统计代写biostatistics代考|The Normal Probability Model

The choice of a probability model for continuous variables is generally based on historical data rather than a particular set of conditions. Just as there are many discrete probability models, there are also many different probability models that can be used to model the distribution of a continuous variable. The most commonly used continuous probability model in statistics is the normal probability model.

The normal probability model is often used to model distributions that are expected to be unimodal and symmetric, and the normal probability model forms the foundation for many of the classical statistical methods used in biostatistics. Moreover, the distribution of many natural phenomena can be modeled very well with the normal distribution. For example, the weights, heights, and IQs of adults are often modeled with normal distributions.

The standard normal, which will be denoted by $Z$, is a normal distribution having mean 0 and standard deviation 1. The standard normal is used as the reference distribution from which the probabilities and percentiles associated with any normal distribution will be determined. The cumulative probabilities for a standard normal are given in Tables A.1 and A.2; because $99.95 \%$ of the standard normal distribution lies between the values $-3.49$ and $3.49$, the standard normal values are only tabulated for $z$ values between $-3.49$ and $3.49$. Thus, when the value of a standard normal, say $z$, is between $-3.49$ and $3.49$, the tabled value for $z$ represents the cumulative probability of $z$, which is $P(Z \leq z)$ and will be denoted by $\Phi(z)$. For values of $z$ below $-3.50, \Phi(z)$ will be taken to be 0 and for values of $z$ above $3.50, \Phi(z)$ will be taken to be 1. Tables A.1 and A.2 can be used to compute all of the probabilities associated with a standard normal.

The values of $z$ are referenced in Tables A.1 and A.2 by writing $z=a . b c$ as $z=a . b+0.0 c$. To locate a value of $z$ in Table A.1 and A.2, first look up the value $a . b$ in the left-most column of the table and then locate $0.0 c$ in the first row of the table. The value cross-referenced by $a . b$ and $0 . c$ in Tables A.1 and A.2 is $\Phi(z)=P(Z \leq z)$. The rules for computing the probabilities for a standard normal are given below.

## 统计代写|生物统计代写biostatistics代考|Z Scores

The result of converting a non-standard normal value, a raw value, to a $Z$-value is a $Z$ score. A $Z$ score is a measure of the relative position a value has within its distribution. In particular, a $Z$ score simply measures how many standard deviations a point is above or below the mean. When a $Z$ score is negative the raw value lies below the mean of its distribution, and when a $Z$ score is positive the raw value lies above the mean. $Z$ scores are unitless measures of relative standing and provide a meaningful measure of relative standing only for mound-shaped distributions. Furthermore, $Z$ scores can be used to compare the relative standing of individuals in two mound-shaped distributions.
Example 2.41
The weights of men and women both follow mound-shaped distributions with different means and standard deviations. In fact, the weight of a male adult in the United States is approximately normal with mean $\mu=180$ and standard deviation $\sigma=30$, and the weight of a female adult in the United States is approximately normal with mean $\mu=145$ and standard deviation $\sigma=15$. Given a male weighing $215 \mathrm{lb}$ and a female weighing $170 \mathrm{lb}$, which individual weighs more relative to their respective population?

The answer to this question can be found by computing the $Z$ scores associated with each of these weights to measure their relative standing. In this case,
$$z_{\text {male }}=\frac{215-180}{30}=1.17$$
and
$$z_{\text {female }}=\frac{170-145}{15}=1.67$$
Since the female’s weight is $1.67$ standard deviations from the mean weight of a female and the male’s weight is $1.17$ standard deviations from the mean weight of a male, relative to their respective populations a female weighing $170 \mathrm{lb}$ is heavier than a male weighing $215 \mathrm{lb}$.

## 统计代写|生物统计代写biostatistics代考|The Binomial Probability Model

p(X)=n!X!(n−X)!pX(1−p)n−X 为了 X=0,1,…,n

• n!X!(n−X)!是有多少种方式X成功n试验，
• n!=n(n−1)(n−2)⋯3⋅2⋅1和0!=1根据定义，
• p是任何一个成功的概率n试验，
• pX是拥有的概率X成功n试验，
• 1−p是任何一个失败的概率n试验，
• (1−p)n−X是得到的概率n−X失败n试验。
图 2.24 给出了二项分布的示例。请注意，当二项分布的右尾较长时p<0.5, 一条较长的尾巴在左边时p>0.5, 并且是对称的p=0.5.

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。