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# System prepended metadata

title: 機率TP9
tags: [Theory of Probability, 必修, NTNU, CSIE]

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title: 機率TP9
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# Theory of Probability<br>Continuous Random Variables: Joint PDFs, Conditioning, Expectation and Independence

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`

## Multiple Continuous Random Variables
- Two continuous random variables $X$ and $X$ associated with a common experiment are jointly continuous and can be described in term of a joint PDF $f_{X,Y}$ satisfying
    - $f_{X,Y}$ is a non-negative function
    - Normalization Probability 
        $\int^\infty_{-\infty}\int^\infty_{-\infty} f_{X,Y}(x,y)dxdy= 1$
- Similarly, $f_{X,Y}(a,c)$ can be viewed as the "probabilty per unit area" in the vicinity of (a,c)
    $$
    P(a\leq X\leq a+\delta,c\leq Y\leq c+\delta)\\
    =\int_a^{a+\delta}\int_c^{c+\delta}f_{X,Y}(x,y)dxdy=f_{X,Y}(a,c)\cdot\delta^2
    $$
    - Where $\delta$ is a small positive number
- Marginal Probability
    $$
    \begin{align}
    P(X\in A)&=P(X\in A \text{ and }Y\in(-\infty,\infty))\\
    &=\int_{X\in A}\int^\infty_{-\infty}f_{X,Y}(x,y)dydx
    \end{align}
    $$
    - We have already defined that
        $$
        P(X\in A)=\int_{X\in A}f_X(x)dx
        $$
    - We thus have the *marginal PDF*
        $$
        f_X(x)=\int^\infty_{-\infty}f_{X,Y}(x,y)dy
        $$
        Similary
        $$
        f_Y(y)=\int^\infty_{-\infty}f_{X,Y}(x,y)dy
        $$
## Joint CDFs
<!-- slide 5 -->


## Conditional Expectation Given an Event
<!-- slide 17 -->
- The conditional expectation of a continuous random variable $X$, given an event $A$ ($P(A)>0$)
- 

## Conditional Expectation Given an Random Variable
<!-- slide 19 -->
- The properties of unconditional expectation carry though, with the obvious modifications, to conditional expectation

## Total Probability/Expectation Theorems
- Total Probability Theorem
    - For any event $A$ and a continuous random variable $Y$
- Total Expectation Theorem
    - For any continuous random variable $X$ and $Y$ 

## Independence
<!--  slide 22 -->
- Two continuous random variables $X$ and $Y$ are independent if
    $$
    f_{X,Y}(x,y) = f_X(x)f_Y(y), \forall x, y
    $$
    - Since that
        $$
        f_{X,Y}(x,y)=f_Y(y)f_{X|Y}(x|y)=f_X(x)f_{Y|X}(y|x)
        $$
        - We therefore have 
            $$
            f_{X|Y}(x,y)=f_X(x),\forall y\text{ if }f_Y(y)>0
            $$
        - Or 

## More Factors about Independence
<!-- Slide 23 -->
- If two continuous random variables $X$ and $$ are independence, then 
    - Any two events of the forms {X \in A} an {Y \in B} are independent
    - It also implies that
    - The converse statement is also true (See the end-of-chapter problem 32)
<!-- Slide 24 -->
- If two continuous rndom variables $X$ and $Y$ are independent, then 
    - $E[XY] = E[X]E[Y]$

## Recall: the Discrete Bayes' Rule
- Let $A_1, A_2, \cdots, A_n$ be disjoint events that form a partition of the sample space, and assume that $P(A_i) \geq 0$ for ll $i$. Then, for any event $B$ such that $P(B) > 0$ we have 
    $$
    \begin{align}
    P(A_i|B)&=\frac{P(A_i)P(B|A_i)}{P(B)}\\
    &=\frac{P(A_i)P(B|A_i)}{\sum_{k=1}^nP(A_k)P(B|A_k)}\\
    &=\frac{P(A_i)P(B|A_i)}{P(A_1)P(B|A_1)+\cdots+P(A_n)P(B|A_n)}
    \end{align}
    $$

## Inference and Continuous Bayes' Rule
- As we have a model odf underlying but unobserved phenomenon, represented by a random variable $X$ with PDF $f_X$, and we make  noisy measurement Y, which is modeled in terms of a conditional PDF $f_{Y|X}$. Once the expermntal value of $Y$ is measured, what information does this provide on the unknown value of $X$?




