Chapter 3 (General Random Variables): Joint PDFs of Multiple Random Variables (联合概率密度)

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

Joint PDF

• We say that two continuous random variables associated with the same experiment are jointly continuous (联合连续的) and can be described in terms of a joint PDF $f_{X,Y}$ if $f_{X,Y}$ is a nonnegative function that satisfies
  $$P((X,Y)\in B)=\int {\int }_{(x,y)\in B}{f}_{X,Y}(x,y)dxdy$$for every subset $B$ of the two-dimensional plane. In the particular case where $B$ is a rectangle of the form $B=\{(x,y)|a\le x\le b,c\le y\le d\}$, we have
  $$P(a\le X\le b,c\le Y\le d)={\int }_c^d{\int }_a^b{f}_{X,Y}(x,y)dxdy$$Furthermore, by letting $B$ be the entire two-dimensional plane, we obtain the normalization property
  $${\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{f}_{X,Y}(x,y)dxdy=1$$
• To interpret the joint PDF, we let $\delta$ be a small positive number and consider the probability of a small rectangle. We have
  $$P(a\le X\le a+\delta ,c\le Y\le c+\delta )={\int }_c^{c+\delta }{\int }_a^{a+\delta }{f}_{X,Y}(x,y)dxdy\approx f_{X,Y}(a,c)\cdot {\delta }^2$$so we can view $f_{X,Y}(a,c)$ as the “probability per unit area” in the vicinity of $(a,c)$.
• The marginal PDF $f_X$ of $X$ is given by
  $$f_X(x)={\int }_{-\infty }^{\infty }{f}_{X,Y}(x,y)dy$$

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Two-Dimensional Uniform PDF

• Let us fix some subset $S$ of the two-dimensional plane. The corresponding uniform joint PDF on $S$ is defined to be
  
• For any set $A\subset S$. the probability that $(X,Y)$ lies in $A$ is
  $$P((X,Y)\in A)=\int {\int }_{(x,y)\in A}{f}_{X,Y}(x,y)dxdy=\frac{areaofA}{areaofS}$$

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Example 3.11. Buffon’s Needle.
A surface is ruled with parallel lines, which are at distance $d$ from each other. Suppose that we throw a needle of length $l$ on the surface at random. What is the probability that the needle will intersect one of the lines?

SOLUTION

• We assume here that $l<d$ so that the needle cannot intersect two lines simultaneously. Let $X$ be the vertical distance from the midpoint of the needle to the nearest of the parallel lines. and let $\theta$ be the acute angle formed by the axis of the needle and the parallel lines. We model the pair of random variables $(X,\Theta )$ with a uniform joint PDF over the rectangular set $\{(x,\theta )|0\le x\le d/2,0\le \theta \le \pi /2\}$, so that
  
  The needle will intersect one of the lines if and only if
  $$X\le \frac{l}{2}sin\Theta$$so the probability of intersection is
  $$\begin{array}{rl}P(X\le \frac{l}{2}sin\Theta )& =\int {\int }_{x\le \frac{l}{2}sin\theta }{f}_{X,\Theta }(x,\theta )dxd\theta \\ & =\frac{4}{\pi d}{\int }_0^{\pi /2}{\int }_0^{(l/2)sin\theta }dxd\theta \\ & =\frac{4}{\pi d}{\int }_0^{\pi /2}(l/2)sin\theta d\theta \\ & =\frac{2l}{\pi d}\end{array}$$

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Problem 16.
Consider the following variant of Buffon’s needle problem, which was investigated by Laplace. A needle of length $l$ is dropped on a plane surface that is partitioned in rectangles by horizontal lines that are $a$ apart and vertical lines that are $b$ apart. Suppose that the needle’s length $l$ satisfies $l<a$ and $l<b$. What is the expected number of rectangle sides crossed by the needle? What is the probability that the needle will cross at least one side of some rectangle?

SOLUTION

• Let $A$ be the event that the needle will cross a horizontal line, and let $B$ be the probability that it will cross a vertical line. From the analysis of Example 3.11, we have that
  $$P(A)=\frac{2l}{\pi a},P(B)=\frac{2l}{\pi b}$$The expected number of crossed lines is
  $$P(A)+P(B)=\frac{2l(a+b)}{\pi ab}$$
• The probability that at least one line will be crossed is
  $$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$Let $X$ (or $Y$) be the distance from the needle’s center to the nearest horizontal (or vertical) line. Let $\Theta$ be the angle formed by the needle’s axis and the horizontal lines. We have
  $$P(A\cap B)=P(X\le \frac{lsin\Theta }{2},Y\le \frac{lcos\Theta }{2})$$We model the triple $(X,Y,\Theta )$ as uniformly distributed over the set of all $(x,y,\theta )$ that satisfy $0\le x\le a/2,0\le y\le b/2$, and $0\le \theta \le \pi /2$. Hence, within this set, we have
  $$f_{X,Y,\Theta }(x,y,\theta )=\frac{8}{\pi ab}$$The probability $P(A\cap B)$ is
  Thus we have
  $$P(A\cup B)=P(A)+P(B)-P(A\cap B)=\frac{2l}{\pi a}+\frac{2l}{\pi b}-\frac{l^2}{\pi ab}=\frac{l}{\pi ab}(2(a+b)-l)$$

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Problem 17. Estimating an expected value by simulation using samples of another random variable.
Let $Y_1,...,Y_n$ be independent random variables drawn from a common and known PDF $f_Y$. Let $S$ be the set of all possible values of $Y_i$, $S=\{y|f_Y(y)>0\}$. Let $X$ be a random variable with known PDF $f_X$, such that $f_X(y)=0$, for all $y\notin S$. Consider the random variable
$$Z=\frac{1}{n}\sum _{i=1}^n{Y}_i\frac{f_X(Y_i)}{f_Y(Y_i)}$$Show that
$$E[Z]=E[X]$$

SOLUTION
We have
$$E[Y_i\frac{f_X(Y_i)}{f_Y(Y_i)}]＝{\int }_{S}y\frac{f_X(y)}{f_Y(y)}{f}_Y(y)dy={\int }_{S}y{f}_X(y)dy=E[X]$$Thus,
$$E[Z]=\frac{1}{n}\sum _{i=1}^{n}E[Y_i\frac{f_X(Y_i)}{f_Y(Y_i)}]=\frac{1}{n}\sum _{i=1}^{n}E[X]=E[X]$$

Joint CDFs

联合分布函数

• If $X$ and $Y$ are two random variables associated with the same experiment, we define their joint CDF by
  $$F_{X,Y}(x,y)=P(X\le x,Y\le y)$$
• In particular, if $X$ and $Y$ are described by a joint PDF $f_{X,Y}$, then
  $$F_{X,Y}(x,y)=P(X\le x,Y\le y)={\int }_{-\infty }^x{\int }_{-\infty }^y{f}_{X,Y}(s,t)dtds$$Conversely, the PDF can be recovered from the CDF by differentiating:
  $$f_{X,Y}(x,y)=\frac{{\partial }^2{F}_{X,Y}(x,y)}{\partial x\partial y}$$

Expectation

$$E[g(X,Y)]={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }g(x,y)f_{X,Y}(x,y)dxdy$$

• As an important special case, for any scalars $a,b$, and $c$, we have
  $$E[aX+bY+x]=aE[X]+bE[Y]+c$$