# 电气工程代写|数字信号过程代写digital signal process代考|ECE513

## 电气工程代写|数字信号过程代写digital signal process代考|Statistics and Random Variables

This section will provide an overview of random variables and the statistical metrics used to quantify their behavior. In simple terms, a random variable maps the outcome of an experiment of chance into numbers. Flipping a coin to get heads or tails can be mapped to a variable $r$ which assumes the numbers 1 for heads and $-1$ for tails. In a game of dice, one of six unique faces will appear after each throw. As soon as we assign the numbers one through six to these unique faces, we may use a random variable to describe the experiment of chance. These random variables may be discrete in nature, as was the case for the coin flipping experiment where only two states are possible or continuous as would be the case for a measurement of the outside air temperature at 7:00 o’clock every morning. A random variable $t$, which represents early morning temperatures, will deviate around an average value in a manner that can be described probabilistically. How probable is it that the measurements end up between $10^{\circ} \mathrm{C}$ and $13^{\circ} \mathrm{C}$, what would be their average and how far from that mean (average) do the values deviate? Finding mathematical descriptions that statistically parameterize random variables is the topic of this section.

A random variable is described by its probability density function, or PDF, which provides information regarding the probability of every possible outcome of our experiment of chance. There are several very important probability density functions that appear in communication systems, including the Gaussian, Uniform, and Raleigh PDF.

A complete description of a random variable as provided by a PDF may not be necessary in our day to day engineering tasks. Most of the time, statistical metrics providing less information like mean, variance, and RMS value are more than adequate. These metrics can either be derived from the PDF directly or computed by observing a large sample set of outcomes in which case we refer to them as sample statistics.

## 电气工程代写|数字信号过程代写digital signal process代考|The Probability Density Function

The PDF or probability density function, $\operatorname{Prob}(x)$, of a random variable $x$ is used to calculate the likelihood of a random variable assuming a value between two limits $x_1$ and $x_2$. To calculate that probability, the PDF must be integrated between these two positions.
$$\text { Probability }\left(x>x_1 \& x<x_2\right)=\int_{x_1}^{x_2} \operatorname{Prob}(x) d x$$
A game of darts can illustrate the common types of probability density functions. Imagine a vertical line running through the center of the dart board, and as we throw our darts we record the displacement, $x$, to the right of that vertical line. When the dart hits to the left of the vertical center line, the displacement we record is negative. The displacement, $x$, in this case is a random variable, which assumes values that are continuous in nature.

The probability density function for this random variable is called a Gaussian PDF [6] also known as the famous bell curve. It makes intuitive sense that the probability density should be higher in the center than further out. The equation for the Gaussian PDF must be integrated between two displacements to find the probability of the dart landing in between them.The equation for the Gaussian PDF is shown below.
$$\operatorname{Prob}(x)=\frac{1}{\sqrt{2 \pi \sigma_x^2}} \exp \operatorname{x\infty }\left(-\frac{\left(x-m_x\right)^2}{2 \sigma_x^2}\right)$$
There are several very important facts that we should mention about the Gaussian PDF.

1. The integral of the equation above has no closed form solution. To find the probability of the dart landing between two displacements, we use numerical methods.
2. The Gaussian PDF is completely defined by two variables, the mean $m$ and the standard deviation $\sigma$. The standard deviation is a measure of uncertainty, or in this case indicates your skill at throwing darts. If you are new to the game of darts, the amount of uncertainty, $\sigma$, in your throws will be greater than that of a professional.
3. The Gaussian PDF is the most important and common probability density function in nature. There are random variables that are themselves made up of the summation of other random variables. The amount of money that a casino makes at the end of the day is really the summation of the amount earned at each table. The Dow Jones industrial average is another great example of a random number being made up of the fluctuations of the value of many different stocks. The central limit theorem states that a random variable that is the summation of other random variables will tend to feature a Gaussian PDF. This is the case even if the constituent random variables were not Gaussian in nature. Since most random events are actually a combination of many different random occurrences, it is no wonder that the Gaussian PDF is so common.

# 数字信号过程代考

## 电气工程代写|数字信号过程代写digital signal process代考|The Probability Density Function

PDF 或概率密度函数， $\operatorname{Prob}(x)$ ，一个随机变量 $x$ 用于计算随机变量假设值介于两个限制之间的可能性 $x_1$ 和 $x_2$. 为了计算该概率， PDF 必须在这两个位置之间进行积分。
Probability $\left(x>x_1 \& x<x_2\right)=\int_{x_1}^{x_2} \operatorname{Prob}(x) d x$

$$\operatorname{Prob}(x)=\frac{1}{\sqrt{2 \pi \sigma_x^2}} \exp \times \infty\left(-\frac{\left(x-m_x\right)^2}{2 \sigma_x^2}\right)$$

1. 上述方程的积分没有封闭形式的解。为了找到两个位移之间落镖的概率，我们使用数值方法。
2. 高斯 PDF 完全由两个变量定义，均值 $m$ 和标准差 $\sigma$. 标准偏差是对不确定性的衡量，或者在这种情况下 表明您投邤飞镖的技能。如果您是飞镖游戏的新手，不确定性的数量， $\sigma$ ，在你的投郑中将比专业的更 大。
3. 高斯 PDF 是自然界中最重要和最常见的概率密度函数。有些随机变量本身是由其他随机变量的总和组成 的。奢场在一天结束时赚到的钱实际上是每张桌子上赚到的钱的总和。道琼斯工业平均指数是由许多不 同股票的价值波动组成的随机数的另一个即好的例子。中心极限定理指出，作为其他随机变量之和的随 机变量将倾向于具有高斯 PDF。即使组成随机变量本质上不是高斯的，情况也是如此。由于大多数随机 事件实际上是许多不同随机事件的组合，难怪高斯 PDF 如此普遍。

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