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# System prepended metadata

title: 機率TP8
tags: [Theory of Probability, 必修, NTNU, CSIE]

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title: 機率TP8
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# Theory of Probability：<br>Continuous Random Variables - Basics

NTNU 機率論

##### [Back to Note Overview](https://reurl.cc/XXeYaE)
##### [Back to Theory of Probability](https://hackmd.io/@NTNUCSIE112/H1OPnkA4v)
###### tags: `NTNU` `CSIE` `必修`  `Theory of Probability`


## Normality is Preserved by Linear Transformations
If $X$ is a normal random variable with mean $\mu$ and variance $\sigma^2$, and if $a(a\neq 0)$ and $b$ are scalars, then the random variable
$$
Y = aX+b
$$
is also normal with mean and variance
$$
E[Y] = a\mu+b\\
\text{var}(Y)=a^2\sigma^2
$$
    

## Standard Normal Random Variable
- A normal random variable $Y$ with zero mean $\mu = 0$ and unit variance $\sigma^2=1$ is said to be a **standard normal**
    $$
    f_Y(y) = {1\over \sqrt{2\pi}}e^{-{y^2\over 2}},\  -\infty \leq y \leq \infty
    $$
- Normalization Property
    $$
    \int^\infty_{-\infty}\frac{1}{\sqrt{2\pi}}e^{-\frac{y^2}{2}}dy=1
    $$
- The standard normal is symmetric around $y=0$

### The PDF of a Random Variable Can be Arbitratily Large
![](https://i.imgur.com/rBs8KCP.png)
- Consider a random variable $X$ with PDF
    $$
    f_X(x)=\begin{cases}
    \frac{1}{2\sqrt{x}},&\text{if }0<x\leq1,\\
    0,&\text{otherwise},
    \end{cases}
    $$
    - The PDF value becomes infinite large as $x$ approaches zero
- Normalizatino Property
    $$
    \int^1_0f_X(x)dx=\int^1_0\frac{1}{2\sqrt{x}}dx=\sqrt{x}|^1_0=1
    $$

## Expectation of a Continuous Random Variable 
- Let $X$ be a continuous random variable with PDF $f_x$
    - The expectaion of $X$ is defined by
        $$
        E[X]=\int^\infty_{-\infty}x\cdot f_X(x)dx
        $$
    - The expectation of a function $g(X)$ has the form
        $$
        E[g(X)]=\int^\infty_{-\infty}g(x)\cdot f_X(x)dx
        $$
    - The variance of $X$ is defuned by
        $$
        \text{var}(X)=E[X^2]-(E[X])^2\geq0
        $$
        - we also have
            $$
            \text{var}(X)=E[X^2]-(E[X])^2\geq0
            $$
- if $Y = aX+b$, where $a$ and $b$ are given scalars, then 
    $$
    E[Y]=aE[X]+b\\
    \text{var}(Y)=a^2\text{var}(X)
    $$

### Expaonential Random Variable
- An **expaonential** random variable $X$ has a PDF if the form
    $$
    f_X(x)=\begin{cases}\lambda e^{-\lambda x},&\text{if }x\geq 0,\\
    0,&\text{otherwise}
    \end{cases}
    $$
- Normalization Property
    $$
    \int_{-\infty}^\infty f_X(x)dx=\int_0^\infty\lambda e^{-\lambda x}dx=-e^{-\lambda x}|^\infty_0=1
    $$
- The probability that $X$ exceeds a certain value decreasese exponentially
    $$
    P(X\geq a)=\int_a^\infty\lambda e^{-\lambda x}dx=e^{-\lambda a}
    $$

## Properties of a CDF
- The CDF $F_X(x)$ is *monotonically(單調) non-decreasing*
    - if $x_i\leq x_j$, then $F_X(x_i)\leq F_X(x_j)$
- The CDF $F_X(x)$ tends to 0 as $x\to-\infty$, ans to 1 as $x\to\infty$
- If $X$ is deicrete, then $F_X(x)$ is a *piecewise constant* function(分段常值函數) of $x$.![](https://i.imgur.com/fJGWDJE.png)
- If $X$ is continuous, then $F_X(x)$ is a *continous*  function of $x$.![](https://i.imgur.com/67OqSz1.png)

- If $X$ is descrete and takes integer values, the PMF and the CDF can be obtained from each other by summing or differencing.
    $$
    \begin{align}
    F_X(k)&=P(X\leq k)&=\sum_{i=-\infty}^kp_X(i),\\
    p_X(k)&=P(X\leq k)-P(X\leq k-1)&=F_X(k)-F_X(k-1)
    \end{align}
    $$
- If X is continous, the PDF and the CDF can be obtained from each other by integration or differentiation
    $$
    F_X(x)=P(X\leq x)=\int_{-\infty}^\infty f_X(t)dt,\\
    f_X(x)=\frac{dF_X(x)}{dx}
    $$
    - The second equality is valid for those $x$ for which the CDF has a derivative (or at which the PDF is continuous)

## Example(slide(TP8) p25)
![](https://i.imgur.com/NWAImQ9.png)
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## CDF of the **Standard Normal**
- The CDF of the standard normal $Y$, denoted as $\phi(y)$, is recorded in a table and is a very useful tool for calculating various probabilities, including normal variables.
    $$
    \phi(y)=P(Y\leq y)=P(Y<y)=\int_{-\infty}^y\frac{1}{\sqrt{2\pi}}e^{\frac{-t^2}{2}}dt
    $$
    - The table only provides the value of $\phi(y)$ for $y\geq 0$
    - Because the symmetry of the PDF, the CDF at negative values of $Y$ can be computed from corresponding positive ones
        $$
        \begin{align}
        \phi(-0.5)&=P(Y\leq -0.5)=1-P(Y\leq 0.5)\\
        &=1-\phi(0.5)=1-0.6915\\
        &=0.3085
        \end{align}
        $$
        > $\phi(-y)=1-\phi(y), \forall y$
## Table of the CDF of **Standard Normal**
![](https://i.imgur.com/L7RXm5Y.png =x500)
## CDF Calculation of the **Normal**
- The CDF of a normal random variable $X$ with mean $\mu$ and variance $\sigma^2$ is obtained using the standard normal table as
    $$
    P(X\leq x)=P\left(\frac{X-\mu}{\sigma}\leq\frac{x-\mu}{\sigma}\right)=P\left(Y\leq\frac{x-\mu}{\sigma}\right)=\phi\left(\frac{x-\mu}{\sigma}\right)
    $$
    - Let $Y=\frac{X-\mu}{\sigma}$, since $X$ is normal and $Y$ is a linear function of $X$, $Y$ hence is also normal (with mean 0 and variance 1).
        $$
        E[Y]=\frac{E[X]-\mu}{\sigma}=0, \text{var}(Y)=\frac{\text{var}(X)}{\sigma^2}=1
        $$

## Relation between the **Geometric** and **Exponential**
- The CDF of the geometric
- The CDF of the exponential
- Compare the above two CDFs and let

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