## 1.10 Independent Random Variables and Conditioning When There Is Dependence

Random variables X_{1}, …, X_{k} are said to be mutually independent if

$\begin{array}{l}\hfill {F}_{{X}_{1},\dots ,{X}_{k}}({x}_{1},\dots ,{x}_{k})=\prod _{i=1}^{k}{F}_{{X}_{i}}({x}_{i})\text{for all}{x}_{1},\dots ,{x}_{k}.\end{array}$

(8a)

Equivalently, for mutually independent rv’s,

$\begin{array}{l}\hfill {f}_{{X}_{1},\dots ,{X}_{k}}({x}_{1},\dots ,{x}_{k})=\prod _{i=1}^{k}{f}_{{X}_{i}}({x}_{i})\text{for all}{x}_{1},\dots ,{x}_{k},\end{array}$

(8b)

holds for their joint pdf or joint pmf. The conditional pmf or pdf of X_{r+1}, …, X_{k} given (X_{1}, …, X_{r}) = (x_{1}, …, x_{r}) when ${f}_{{X}_{1},\dots ,{X}_{r}}({x}_{1},\dots ,{x}_{r})>0$ is

$\begin{array}{l}\hfill {f}_{({X}_{r+1},\dots}\end{array}$

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