University of Alberta Guide:STAT:222:Independence and Conditional Expectations

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Notes

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 $(P (X_{1} \leq x_{1}, . . ., X_{d} \leq x_{d}))$ can be multiplied together.

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

See the equations section for some more examples.

Need Info for Conditional Exepctation and Conditional Functions

Equations

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

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University of Alberta Guide:STAT:222:Independence and Conditional Expectations

Notes

Equations

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