Chapter 3 (General Random Variables): Normal Random Variables (正态随机变量)

本文为 $Introduction$ $to$ $Probability$ 的读书笔记

Normal Random Variables

• A continuous random variable $X$ is said to be normal or Gaussian if it has a PDF of the form
  $$f_X(x)=\frac{1}{\sqrt{2\pi }\sigma }{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}$$where $\mu$ and $\sigma$ are two scalar parameters characterizing the PDF, with $\sigma$ assumed positive. It can be verified that the normalization property holds
  $$\frac{1}{\sqrt{2\pi }\sigma }{\int }_{-\infty }^{\infty }{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}dx=1$$
  
• The mean and the variance can be calculated to be
  $$\begin{gathered} E[X]=\mu \\ var(X)={\sigma }^2 \end{gathered}$$To see this, note that the PDF is symmetric around $\mu$, so the mean can only be $\mu$. Furthermore, the variance is given by
  $$var(X)=\frac{1}{\sqrt{2\pi }\sigma }{\int }_{-\infty }^{\infty }(x-\mu {)}^2{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}dx$$Using the change of variables $y=(x-\mu )/\sigma$ and integration by parts, we have
  $$\begin{array}{rl}var(X)& =\frac{{\sigma }^2}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }{y}^2{e}^{-\frac{y^2}{2}}dy\\ & =\frac{{\sigma }^2}{\sqrt{2\pi }}(-y{e}^{-y^2/2}){|}_{-\infty }^{\infty }+\frac{{\sigma }^2}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }{e}^{-y^2/2}dy\\ & =\frac{{\sigma }^2}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }{e}^{-y^2/2}dy\\ & ={\sigma }^2(normalizationpropertyofthenormalPDF)\end{array}$$
  
  

The Standard Normal Random Variable

标准正态随机变量

• A normal random variable $Y$ with zero mean and unit variance is said to be a standard normal. Its CDF is denoted by $\Phi$:
  $$\Phi (y)=P(Y\le y)=P(Y<y)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^y{e}^{-\frac{t^2}{2}}dt$$
• It is recorded in a table
  

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• Let $X$ be a normal random variable with mean $\mu$ and variance ${\sigma }^2$. We “standardize” $X$ by defining a new random variable $Y$ given by
  $$Y=\frac{X-\mu }{\sigma }$$Since $Y$ is a linear function of $X$, it is normal. Furthermore,
  $$\begin{gathered} E[Y]=\frac{E[X]-\mu }{\sigma }=0 \\ var(Y)=\frac{var(X)}{{\sigma }^2}=1 \end{gathered}$$Thus, $Y$ is a standard normal random variable.
• This fact allows us to calculate the probability of any event defined in terms of $X$: we redefine the event in terms of $Y$, and then use the standard normal table.
  

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• Normal random variables play an important role in a broad range of probabilistic models. Mathematically, the key fact is that the sum of a large number of independent and identically distributed (not necessarily normal) random variables has an approximately normal CDF. This property is captured in the celebrated $central$ $limit$ $theorem$. (中心极限定理)
  • Normal random variables are often used in signal processing and communications engineering to model noise and unpredictable distortions of signals.

Example 3.8. Signal Detection.
A binary message is transmitted as a signal $s$, which is either $-1$ or $+1$. The communication channel corrupts the transmission with additive normal noise with mean $\mu =0$ and variance ${\sigma }^2$ . The receiver concludes that the signal $-1$ (or $+1$) was transmitted if the value received is $<0$ (or $\ge 0$), respectively). What is the probability of error?

SOLUTION

• An error occurs whenever $-1$ is transmitted and the noise $N$ is at least $1$, or whenever $+1$ is transmitted and the noise $N$ is smaller than $-1$. In the former case. the probability of error is
  $$\begin{array}{rl}P(N\ge 1)& =1-P(N<1)=1-P(\frac{N-\mu }{\sigma }<\frac{1-\mu }{\sigma })\\ & =1-\Phi (\frac{1-\mu }{\sigma })=1-\Phi (\frac{1}{\sigma })\end{array}$$In the latter case. the probability of error is the same, by symmetry.