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239 joint cumulative (probability) distribution function of $X$ and $Y.$ \[ F\left(x,y\right) =\Pr{\left\{X\le x\ \mathrm{and}\ Y\le y\right\}},\\ \mathrm{where} -\infty\lt x,\ y\lt \infty \]
247 joint cumulative (probability) distribution function of $X_1,\ldots,X_n.$ \[ F\left( x_1, \ldots, x_n \right) =\Pr{ \left\{ X_1 \le x_1\ \mathrm{and} \ldots\mathrm{and} \ X_n \le x_n \right\} },\\ \mathrm{where} -\infty \lt x_1, \ldots, x_n \lt \infty \]
239 cumulative distribution functions of $X$ and $Y$ given that $F$ is the joint cumulative probability distribution function of $X$ and $Y.$ $F_x$ and $F_y$ are sometimes called the marginal distributions of $X$ and $Y$. \[ F_X\left(x\right) =\lim\limits_{y\rightarrow\infty}{F(x,y)} =F\left(x,\infty\right) \\ F_Y\left(y\right) =\lim\limits_{x\rightarrow\infty}{F(x,y)} =F\left(\infty,y\right) \]
240 joint probability mass function of $X$ and $Y$ \[ \Pr{\left(x,y\right)}=\Pr{\left\{X=x\ \mathrm{and}\ Y=y\right\}} \]
240 probability mass functions of discrete random variables $X$ and $Y$ given that $p$ is their joint probability mass function. The individual probability mass functions $p_x(x)$ and $p_y(y)$ are sometimes called marginal probability mass functions of $X$ and $Y.$ \[ p_X\left(x\right) =\Pr{\left\{X=x\right\}} =\sum_{y:p\left(x,y\right)\gt 0}{p(x,y)} \\ p_Y\left(y\right) =\Pr{\left\{Y=y\right\}} =\sum_{x:p\left(x,y\right)\gt 0}{p(x,y)} \]
242 Let $A$ and $B$ be sets of real numbers and $C=\left\{\left(x,y\right):x\in A,\ y\in B\right\}.$ $X$ and $Y$ are jointly continuous if there exists a function $f\left(x,y\right)$ called the joint probability density function of $X$ and $Y$ defined for all real $x$ and $y$ having the property that for every set $C$ of pairs of real numbers (that is, $C$ is a set in the two-dimensional plane), the given equation holds. \[ \eqalign{ \Pr{\left\{\left(X,Y\right)\in C\right\}} &=\Pr{\left\{X\in A,Y\in B\right\}}\\ &=\iint\limits_{\left(x,y\right)\in C} f\left(x,y\right)dx\,dy\\ &=\int\limits_{B}\int\limits_{A} f\left(x,y\right)dx\,dy } \]
247 More generally, let $A_1,\ldots,A_n$ be $n$ sets of real numbers and $C=\left\{\left(x_1,\ldots,x_n\right):x_1\in A_1,\ldots,x_n\in A_n\right\}.$ Then the $n$ random variables $X_1,\ldots,X_n$ are jointly continuous if there exists a function $f\left(x_1,\ldots,x_n\right),$ called the joint probability density function, such that for any set $C$ in $n\textrm{-space},$ the given equation holds. \[ \eqalign{ \Pr{\left\{\left(X_1,\ldots,X_n\right)\in C\right\}} &= \Pr{\left\{X_1\in A_1,\ldots,\ X_n\in A_n\right\}}\\ &= \idotsint\limits_{x_1, \ldots, x_n \in C} f\left(x_1, \ldots, x_n\right) dx_1\cdots dx_n\\ &= \int\limits_{A_1}\cdots\int\limits_{A_n}{f\left(x_1,\ldots,x_n\right)dx_1\cdots dx_n} } \]
242 \[ \eqalign{ F\left(x,y\right) &=\Pr{\left\{X\le x\land Y\le y\right\}} \\ &=\int\limits_{-\infty}^{y}\int\limits_{-\infty}^{x}f\left(x,y\right)dx\,dy } \]
242 \[ f\left(x,y\right)=\frac{\partial^2}{\partial x\partial y}F\left(x,y\right) \]
243 If $X$ and $Y$ are jointly continuous with joint probability density function $f,$ then they are individually continuous. Let $F$ denote their joint cumulative probability distribution function, and $f_x\left(x\right)$ and $f_y\left(y\right)$ their individual probability density functions. Then the given equations hold. Functions $f_x\left(x\right)$ and $f_y\left(y\right)$ are sometimes called marginal probability density functions. \[ f_X\left(x\right)=\int\limits_{-\infty}^{\infty}f\left(x,y\right)dy \\ f_Y\left(y\right)=\int\limits_{-\infty}^{\infty}f\left(x,y\right)dx \]
248 Random variables $X$ and $Y$ are independent if for any two sets of real numbers $A$ and $B$ the first equation holds. Equivalently, $X$ and $Y$ are independent if for all $x$ and $y$ the second equation holds. Equivalently, $X$ and $Y$ are independent if the events $E_A=\left\{X\in A\right\}$ and $E_B=\left\{X\in B\right\}$ are independent. If random variables are not independent, they are dependent. \[ \eqalign{ \Pr{\left\{ X \in A, Y \in B \right\}} &= \Pr{\left\{ X \in A \right\}} \Pr{\left\{ Y \in B \right\}} F\left( x, y \right) \\ &= F_X\left( x \right) F_Y\left( y \right) } \]
254 More generally, the $n$ random variables $X_1,\ldots,X_n$ are independent if, for all sets of real numbers $A_1,\ldots,A_n$ the given equation holds. An infinite collection of random variables is independent if every finite subcollection of them is independent. \[ \eqalign{ \Pr{\left\{X_1\in A_1,\ldots,X_n\in A_n\right\}} &=\Pr{\left\{X_1\le x_1,\ldots,X_n\le x_n\right\}} \\ &=\prod_{i=1}^{n}\Pr{\left\{X_i\in A_i\right\}} \\ &=\prod_{i=1}^{n}\Pr{\left\{X_i\in A_i\right\}} } \]
248 If $X$ and $Y$ are discrete random variables, then $X$ and $Y$ are independent if for all $x$ and $y$ the given equation holds. \[ p\left(x,y\right)=p_X\left(x\right)p_Y\left(y\right) \]
248 If $X$ and $Y$ are jointly continuous, then $X$ and $Y$ are independent if for all $x$ and $y$ the given equation holds. \[ f\left(x,y\right)=f_X\left(x\right)f_Y\left(y\right) \]
253 The continuous (discrete) random variables $X$ and $Y$ are independent if and only if their joint probability density (mass) function can be expressed by the given equation. (Note. The author uses $f(x,y)$ and $f_{X,Y}\left(x,y\right)$ interchangeably, the latter stressing the random variables for which $f$ is a density function.) \[ f_{X,Y}\left(x,y\right) = h\left(x\right)g\left(y\right), \\ \mathrm{where} -\infty\lt x,y\lt \infty \]
260 \[ \eqalign{ F_{X+Y}\left(a\right) &= \Pr{\left\{X+Y\le a\right\}} \\ &= \iint\limits_{x+y\le a}{f_X\left(x\right)f_Y\left(y\right)dx\,dy} } \]
261 \[ F_{X+Y}\left(a\right)=\frac{d}{da}F_{X+Y}\left(a\right) \]
260 sums of independent random variables
261 sum of two independent uniform random variable
263 chi-squared with n degrees of freedom \[ \]
267 sums of independent Poisson random variables
267 sums of independent binomial random variables
268 If $X$ and $Y$ are discrete random variables, then the conditional probability mass function of $X$ given that $Y=y$ is defined for all $y:p_Y\left(y\right)\gt 0$ as given. \[ \eqalign{ P_{X \mid Y}\left(x \mid y\right) &= \Pr{\left\{X=x \mid Y=y\right\}} \\ &= \frac{p\left(x,y\right)}{p_Y\left(y\right)} } \]
269 If $X$ and $Y$ are discrete random variables, then the conditional probability distribution function of $X$ given that $Y=y$ is defined for all $y:p_Y\left(y\right)\gt 0$ as given. \[ \eqalign{ F_{X \mid Y}\left(x \mid y\right) &= \Pr{\left\{X\le x \mid Y=y\right\}} \\ &= \sum_{a\le x}{p_{X \mid Y}\left(a \mid y\right)} \\ &= \sum_{a\le x}\frac{p\left(x,y\right)}{p_Y\left(y\right)} } \]
269 If $X$ is independent of $Y,$ then the equations hold. \[ F_{X \mid Y}\left(x \mid y\right)=F_X(x) \\ p_{X \mid Y}\left(x \mid y\right)=p_X\left(x\right) \]
270 If $X$ and $Y$ are continuous random variables, then the conditional probability density function of $X$ given that $Y=y$ is defined for all $y:p_Y\left(y\right)\gt 0$ as given. \[ f_{X \mid Y}\left(x \mid y\right)=\frac{f\left(x,y\right)}{f_Y\left(y\right)} \]
271 If $X$ and $Y$ are continuous random variables, then the conditional cumulative distribution function of $X$ given that $Y=y$ is defined for all $y:p_Y\left(y\right)\gt 0$ as given. \[ \eqalign{ F_{X \mid Y}\left(x \mid y\right) &=\Pr{\left\{X\le x \mid Y=y\right\}}\\ &=\int_{-\infty}^{\infty}{f_{X \mid Y}\left(x \mid y\right)dx} } \]