# University of Alberta Guide/STAT/222/Independence and Conditional Expectations

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The basic idea behind independence is that if your random variables (or vectors) are independent then the combination of several of these random variable/vectors ${\displaystyle \left(P\left(X_{1}\leq x_{1},...,X_{d}\leq x_{d}\right)\right)}$ can be multiplied together.

Given ${\displaystyle T_{1},T_{2}\,}$ are independent ${\displaystyle \lambda \,{\mbox{-Exponential}}}$ random variables, then ${\displaystyle E\left[T_{1}+T_{2}\right]=E\left[2T_{1}\right]=E\left[2T_{2}\right]}$. But ${\displaystyle {\mbox{Var}}\left(T_{1}+T_{2}\right)<{\mbox{Var}}\left(2T_{1}\right)}$. This is because ${\displaystyle {\mbox{Var}}\left(T_{1}+T_{2}\right)={\mbox{Var}}\left(T_{1}\right)+{\mbox{Var}}\left(T_{2}\right)=2{\mbox{Var}}\left(T_{1}\right)}$ by independence, whereas ${\displaystyle {\mbox{Var}}\left(2T_{1}\right)=2^{2}\cdot {\mbox{Var}}\left(T_{1}\right)}$.

See the equations section for some more examples.

## Equations

 ${\displaystyle P\left(X_{1}\leq x_{1},...,X_{d}\leq x_{d}\right)\,}$ ${\displaystyle =P\left(X_{1}\leq x_{1}\right)\cdots P\left(X_{d}\leq x_{d}\right)\,}$ ${\displaystyle F_{X_{1},X_{2},...X_{d}}(x_{1...d})\,}$ ${\displaystyle =F_{X_{1}}(x_{1})\cdots F_{X_{d}}(x_{d})\,}$ ${\displaystyle f_{X_{1},X_{2},...X_{d}}(x_{1...d})\,}$ ${\displaystyle =f_{X_{1}}(x_{1})\cdots f_{X_{d}}(x_{d})\,}$ ${\displaystyle E\left[X_{1}\cdot X_{2}\cdots X_{d}\right]\,}$ ${\displaystyle =E\left[X_{1}\right]\cdots E\left[X_{d}\right]\,}$ ${\displaystyle E\left[g_{1}\left(X_{1}\right)\cdot g_{2}\left(X_{2}\right)\cdots g_{d}\left(X_{d}\right)\right]\,}$ ${\displaystyle =E\left[g_{1}\left(X_{1}\right)\right]\cdots E\left[g_{d}\left(X_{d}\right)\right]\,}$ ${\displaystyle M_{X_{1}+...+X_{d}}(x)\,}$ ${\displaystyle =M_{X_{1}}(x)\cdots M_{X_{1}}(x)\,}$ ${\displaystyle F_{XY}(x,y)\,}$ ${\displaystyle =???\,}$ ${\displaystyle f_{XY}(x,y)\,}$ ${\displaystyle =???\,}$ ${\displaystyle F_{X|Y}(x|y)\,}$ ${\displaystyle =???\,}$ ${\displaystyle F_{X+Y}(x)\,}$ ${\displaystyle =P\left(X+Y\leq x\right)\,}$