Function of several random variables

Summary

Function of two random variables

• Given:
• An experiment
• A sample space \(S\) and events
• A probability measure \(P \) assigning a probability to each event. \(P\) satisfies the probability rules.
• Two real-valued random variables \(X\) and \(Y\) .
• Let \(g:R^2→R\) be an arbitrary function of two variables. Then we can define a new random variable \(Z=g\left( X,Y \right)\) .

Example

• Experiment that picks an outcome from \(S=\{ a,b,c \}\) . Each outcome is equally likely .
• \(X:S→R\) defined by \(X\left( a \right)=1\) , \(X\left( b \right)=2\) and \(X\left( c \right)=3\)
• \(Y:S→R\) defined by \(Y\left( a \right)=-1\) , \(Y\left( b \right)=0\) and \(Y\left( c \right)=1\)
• \(g:R^2→R\) defined by \(g\left( x,y \right)=xy\) and \(Z=g\left( X,Y \right)\)
• Then \(Z\left( a \right)=1⋅\left( -1 \right)=-1\) , \(Z\left( b \right)=2⋅0=0\) and \(Z\left( c \right)=3⋅1=3\)

Function of several random variables

• If we have \(n\) random variables \(X_1,…,X_n\) and \(g:R^n→R\) is a function of \(n\) variables then \(Z=g\left( X_1,…,X_n \right)\) will be a new random variable.