Conditioning on an Event
\[\mathbb{E}[X\lvert A]=\int xf_{X\lvert A}(x)\,\mathrm{d}x\] \[\mathbb{E}[g(X)\lvert A]=\int g(x)f_{X\lvert A}\,\mathrm{d}x\]Total Probability Theorem
\[f_X(x)=\mathbf{P}(A_1)f_{X\lvert A_1}(x)+\ldots+\mathbf{P}(A_n)f_{X\lvert A_n}(x)\]Conditioning on another Random Variable
If \(f_Y(y)>0,\)
\[f_{X\lvert Y}(x\lvert y)=\dfrac{f_{X,Y}(x,y)}{f_Y(y)}\]Independence
\[f_{X,Y}(x,y)=f_X(x)f_Y(y)\]If \(X\) and \(Y\) are independent,
\[\mathbb{E}[XY]=\mathbb{E}[X]\,\mathbb{E}[Y]\]