Let $(X,Y)$ be a discrete bivariate random vector with joint pmf $f(x,y)$ and marginal pmfs $f_X(x)$ and $f_Y(y).$ For any $x$ such that $P(X=x)=f_X(x) >0,$ the conditional pmf of $Y$ given that $X=x$ is the function of $y$ denoted by $f(y\vert x)$ and defined by \[f(y|x) = P(Y=y | X=x) = \frac{f(x,y)}{f_X(x)}.\] For any $y$ such that $P(Y=y)=f_Y(y) >0,$ the conditional pmf of $X$ given that $Y=y$ is the function of $x$ denoted by $f(x\vert y)$ and defined by \[f(x|y) = P(X=x | Y=y) = \frac{f(x,y)}{f_Y(y)}.\]
Let $(X|Y)$ be a continuous bivariate random vector with joint pdf $f(x,y)$ and marginal pdfs $f_X(x)$ and $f_Y(y).$ For any $x$ such that $f_X(x)>0,$ the conditional pdf of $Y$ given that $X=x$ is the function of $y$ denoted by $f(y|x)$ and defined by \[f(y|x) = \frac{f(x,y)}{f_Y(y)},\] For any $y$ such that $f_Y(y)>0,$ the conditional pdf of $X$ given that $Y=y$ is the function of $x$ denoted by $f(x|y)$ and defined by \[f(x|y) = \frac{f(x,y)}{f_X(x)},\]
If $g(Y)$ is a function of $Y$, then the conditional expected value of $g(Y)$ given that $X=x$ is denoted by $E(g(Y)|x)$ and is given by \[E(g(Y)|x) = \sum_y g(y)f(y|x) \:\: \text{ and } \:\: E(g(Y)|x) = \int_{-\infty}^{\infty} g(y)f(y|x)\,dy\] in the discrete and continuous cases, respectively.
The variance of the probability distribution described by $f(y|x)$ is called the conditional variance of $Y$ given $X=x$ and given by \[Var(Y|x) = E(Y^2|x)-(E(Y|x))^2. \]
Let $X(,Y)$ be a bivariate random vector with joint pdf or pmf $f(x,y)$ and marginal pdf or pmfs $f_X(x)$ and $f_Y(y).$ Then $X$ and $Y$ are called independent random variables if, for every $x,y\in\R,$ \[f(x,y) = f_X(x)f_Y(y).\]
If $X$ and $Y$ are independent, the conditional pdf of $Y$ given $X=x$ is
\begin{align*} f(y|x) &= \frac{f(x,y)}{f_X(x)} \\ &= \frac{f_X(x)f_Y(y)}{f_X(x)} \\ &= f_Y(y), \end{align*} regardless of the value of $x.$
Lemma 4.2.1 Let $(X,Y)$ be a bivariate random vector with joint pdf or pmf $f(x,y).$ Then $X$ and $Y$ are independent random variables if and only if there exist functions $g(x)$ and $h(y)$ such that, for every $x,y\in\R,$ \[f(x,y) = g(x)h(y).\]
Theorem 4.2.2
Let $X$ and $Y$ be independent random variables.
a) For any $A\subset\R$ and $B\subset\R,\; P(X\in A,Y\in B)=P(X\in A)P(Y\in B)$; that is, the events $\{X\in A\}$ and $\{Y\in B\}$ are independent events.
b) Let $g(x)$ be a function only of $x$ and $h(y)$ be a function only of $y.$ Then
\[E(g(X)h(Y)) = (Eg(X))(Eh(Y)).\]
Theorem 4.2.3 Let $X$ and $Y$ be independent random variables with moment generating functions $M_X(t)$ and $M_Y(t).$ Then the moment generating function of the random variable $Z=X+Y$ is given by \[M_Z(t) = M_X(t)M_Y(t)\]
Theorem 4.2.4 Let $X\sim N(\mu,\sigma)$ and $Y\sim N(\gamma, \tau^2)$ be independent normal random variables. Then the random variable $Z=X+Y$ has a $N(\mu + \gamma, \sigma^2 + \tau^2)$ distribution.