機率TP9 - HackMD

--- title: 機率TP9 --- # 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  ## Conditional Expectation Given an Event  - The conditional expectation of a continuous random variable $X$, given an event $A$ ($P(A)>0$) - ## Conditional Expectation Given an Random Variable  - 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  - 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  - If two continuous random variables $X$ and $$ are independence, then - Any two events of the forms {X \in A} an {Y \in B} are independent - It also implies that - The converse statement is also true (See the end-of-chapter problem 32)  - If two continuous rndom variables $X$ and $Y$ are independent, then - $E[XY] = E[X]E[Y]$ ## Recall: the Discrete Bayes' Rule - Let $A_1, A_2, \cdots, A_n$ be disjoint events that form a partition of the sample space, and assume that $P(A_i) \geq 0$ for ll $i$. Then, for any event $B$ such that $P(B) > 0$ we have $$ \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 $X$?