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# 数学代写|微积分代写Calculus代考|MTH-211

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## 数学代写|微积分代写Calculus代考|Probability Distributions

A probability distribution describes the probabilistic behavior of a random variable. Our chief interest is in probability distributions associated with continuous random variables, but to gain some perspective we first consider a distribution for a discrete random variable.

$\begin{array}{lcccc}\text { Value of } \mathbf{X} & 0 & 1 & 2 & 3 \ \text { Frequency } & 1 & 3 & 3 & 1 \ \boldsymbol{P}(\boldsymbol{X}) & 1 / 8 & 3 / 8 & 3 / 8 & 1 / 8\end{array}$
We display this information in a probability bar graph of the discrete random variable $X$, as shown in Figure 8.21. The values of $X$ are portrayed by intervals of length 1 on the $x$-axis so the area of each bar in the graph is the probability of the corresponding outcome. For instance, the probability that exactly two heads occurs in the three tosses of the coin is the area of the bar associated with the value $X=2$, which is $3 / 8$. Similarly, the probability that two or more heads occurs is the sum of areas of the bars associated with the values $X=2$ and $X=3$, or $4 / 8$. The probability that either zero or three heads occurs is $\frac{1}{8}+\frac{1}{8}=\frac{1}{4}$, and so forth. Note that the total area of all the bars in the graph is 1 , which is the sum of all the probabilities for $X$.

With a continuous random variable, even when the outcomes are equally likely, we cannot simply count the number of outcomes in the sample space or the frequencies of outcomes that lead to a specific value of $X$. In fact, the probability that $X$ takes on any particular one of its values is zero. What is meaningful to ask is how probable it is that the random variable takes on a value within some specified interval of values.

We capture the information we need about the probabilities of $X$ in a function whose graph behaves much like the bar graph in Figure 8.21. That is, we take a nonnegative function $f$ defined over the range of the random variable with the property that the total area beneath the graph of $f$ is 1 . The probability that a value of the random variable $X$ lies within some specified interval $[c, d]$ is then the area under the graph of $f$ over that interval. The following definition assumes the range of the continuous random variable $X$ is any real value, but the definition is general enough to account for random variables having a range of finite length.

## 数学代写|微积分代写Calculus代考|Exponentially Decreasing Distributions

The distribution in Example 3 is called an exponentially decreasing probability density function. These probability density functions always take on the form
$$f(x)= \begin{cases}0 & \text { if } x<0 \ c e^{-c x} & \text { if } x \geq 0\end{cases}$$
(see Exercise 23). Exponential density functions can provide models for describing random variables such as the lifetimes of light bulbs, radioactive particles, tooth crowns, and many kinds of electronic components. They also model the amount of time until some specific event occurs, such as the time until a pollinator arrives at a flower, the arrival times of a bus at a stop, the time between individuals joining a queue, the waiting time between phone calls at a help desk, and even the lengths of the phone calls themselves. A graph of an exponential density function is shown in Figure 8.23.

Random variables with exponential distributions are memoryless. If we think of $X$ as describing the lifetime of some object, then the probability that the object survives for at least $s+t$ hours, given that it has survived $t$ hours, is the same as the initial probability that it survives for at least $s$ hours. For instance, the current age $t$ of a radioactive particle does not change the probability that it will survive for at least another time period of length $s$. Sometimes the exponential distribution is used as a model when the memoryless principle is violated, because it provides reasonable approximations that are good enough for their intended use. For instance, this might be the case when predicting the lifetime of an artificial hip replacement or heart valve for a particular patient. Here is an application illustrating the exponential distribution.

## 数学代写|微积分代写Calculus代考|Probability Distributions

$\begin{array}{lcccc}\text { Value of } \mathbf{X} & 0 & 1 & 2 & 3 \ \text { Frequency } & 1 & 3 & 3 & 1 \ \boldsymbol{P}(\boldsymbol{X}) & 1 / 8 & 3 / 8 & 3 / 8 & 1 / 8\end{array}$

## 数学代写|微积分代写Calculus代考|Exponentially Decreasing Distributions

$$f(x)= \begin{cases}0 & \text { if } x<0 \ c e^{-c x} & \text { if } x \geq 0\end{cases}$$
(参见练习23)。指数密度函数可以为描述随机变量提供模型，例如灯泡、放射性粒子、牙冠和许多电子元件的寿命。他们还对某些特定事件发生之前的时间进行建模，例如传粉者到达花朵的时间，公共汽车到达车站的时间，个体加入队列的时间，呼叫服务台的等待时间，甚至电话本身的长度。指数密度函数的曲线图如图8.23所示。

## 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。