Conditional expectation and conditional variance, introduction

Summary

Conditional expectation and conditional variance

• If \(X\) and \(Y\) are two random variables then

\[E\left( Y | X \right)\]

• is the conditional expectation of \(Y\) given \(X\) and

\[Var\left( Y | X \right)\]

• is the conditional variance of \(Y\) given \(X\) .

Example:

• Toss two dice.
• Let \(Y_1\) be the outcome of the first, \(Y_2\) be the outcome of the second and \(X\) be the sum.
• \(E\left( Y_1 \right)=3.5\) but \(E\left( Y_1 | X \right)\) is the expected value of \(Y_1\) given that I know the sum .
• For example, if \(X=2\) it must be the case that \(Y_1=1\) and \(E\left(Y_1 | X=2 \right)=1\) .
• Similarly, \(E\left( Y_1 | X=12 \right)=6\) ( \(Y_1\) must be 6) and \(E\left( Y_1 | X=3 \right)=1.5\) (there are two ways that \(X=3\) : \(Y_1=1\) and \(Y_2=2\) or \(Y_1=2\) and \(Y_2=1\) and these are equally likely).

About conditional moments

• Once \(X\) is observed, \(E\left(Y | X \right)\) is a number. Before \(X\) is observed, \(E\left( Y | X \right)\) is a random variable.
• If \(X,Y\) are independent random variables, then

\[E\left( Y | X \right)=E(Y) \textrm{ and } Var\left( Y| X \right)=Var(Y)\]

• If \(X,Y\) is a random variable and \(a,b\) are constants, then

\[E\left( a+bY|X \right)=a+bE\left( Y|X \right)\]

\[Var\left( a+bY|X \right)=b^2Var(Y|X)\]