Functions of Random Variables¶
Linear Function of a Discrete Random Variable¶
\[Y=aX+b\]
\[p_Y(y)=p_X\!\left(\frac{y-b}{a}\right)\]
Linear Function of a Continuous Random Variable¶
\[Y=aX+b\]
\[f_Y(y)=\dfrac{1}{\lvert a\rvert}f_X\!\left(\frac{y-b}{a}\right)\]
General Function \(g(X)\) of a Continuous Random Variable¶
Two steps: * Find the CDF of \(Y\) * Differentiate
Functions of Multiple Random Variables¶
Now, consider a function of two random variables \(g(X,Y)\). It is also a random variable, i.e., \(g(X,Y)=Z\) with PMF
\[p_Z(z)=\mathbf{P}(g(X,Y)=z)=\sum_{(x,y):\,g(x,y)=z}p_{X,Y}(x,y).\]
The latter states that we have to sum the joint probability mass function over all pairs \((x,y)\) such that \(g(x,y)=z\). Its expectation is
\[\mathbb{E}[g(X,Y)]=\sum_{x}\sum_{y}g(x,y)p_{X,Y}(x,y).\]