Expected value and variance of a chi-square k

Problem

  1. \(Y ~ χ_k^2\) . Show that \(E\left( Y \right)=k\) . Hint: If \(Y ~ χ_k^2\) then we can write \(Y=Z_1^2+Z_2^2+ \ldots +Z_k^2\) where \(Z_1,Z_2, \ldots ,Z_k\) are \(n\) independent random variables and \(Z_j \sim N\left( 0,1 \right), j=1, \ldots ,k\) .
  2. Show that \(Var\left( Y \right)=2k\) .

Solution

a. \(Y=Z_1^2+Z_2^2+ \ldots +Z_k^2\) so

\[E\left( Y \right)=E\left( Z_1^2+Z_2^2+ \ldots +Z_k^2 \right)=E\left( Z_1^2 \right)+E\left( Z_2^2 \right)+ \ldots +E\left( Z_k^2 \right)\]

Since each \(Z_j\) is standard normal, \(E\left( Z_j^2 \right)=1\) and \(E\left( Y \right)=1+1+ \ldots +1=k\) .

b.

\[Var\left( Y \right)=Var\left( Z_1^2+Z_2^2+ \ldots +Z_k^2 \right)=Var\left( Z_1^2 \right)+Var\left( Z_2^2 \right)+ \ldots +Var\left( Z_k^2 \right)\]

Since each \(Z_j\) is standard normal, \(Var\left( Z_j^2 \right)=2\) and \(Var\left( Y \right)=2+2+ \ldots +2=2k\) .