Function of a random variable

Summary

• Given:
• An experiment
• A sample space \(S\) and events
• A probability measure \(P \) assigning a probability to each event. \(P\) satisfies the probability rules.
• A random variable \(X\)
• Let \(g:R→R\) be an arbitrary function. Then we can define a new random variable

\[Y=g\left( X \right)\]

• as the composition of \(g\) and \(X\) .

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 \(X\left( c \right)=3\)
• \(g:R→R\) defined by \(g\left( x \right)=x^2\) and \(Y=g\left( X \right)\)
• Then \(Y\left( a \right)=1^2=1\) , \(Y\left( b \right)=2^2=4\) and \(Y\left( c \right)=3^2=9\)