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

Random variables X1, …, Xk are said to be mutually independent if

$\begin{array}{l}\hfill {F}_{{X}_{1},\dots ,{X}_{k}}\left({x}_{1},\dots ,{x}_{k}\right)=\prod _{i=1}^{k}{F}_{{X}_{i}}\left({x}_{i}\right)\text{for all}{x}_{1},\dots ,{x}_{k}.\end{array}$ (8a)

Equivalently, for mutually independent rv’s,

$\begin{array}{l}\hfill {f}_{{X}_{1},\dots ,{X}_{k}}\left({x}_{1},\dots ,{x}_{k}\right)=\prod _{i=1}^{k}{f}_{{X}_{i}}\left({x}_{i}\right)\text{for all}{x}_{1},\dots ,{x}_{k},\end{array}$ (8b)

holds for their joint pdf or joint pmf. The conditional pmf or pdf of Xr+1, …, Xk given (X1, …, Xr) = (x1, …, xr) when ${f}_{{X}_{1},\dots ,{X}_{r}}\left({x}_{1},\dots ,{x}_{r}\right)>0$ is

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

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