Expected value and variance of a chi-square 1

Problem

  1. \(Z ~ N(0,1)\) . Show that \(E\left( Z^2 \right)=1\) . Hint: \(Var\left( Z \right)=E\left( Z^2 \right)-E{\left( Z \right)}^2\) , see Problem: Alternative definitions of Var(X)
  2. \(Y ~ χ_1^2\) . Use a. to show that \(E\left( Y \right)=1\) .
  3. If \(Z ~ N(0,1)\) then one can show that \(E\left( Z^4 \right)=3\) (this is called the kurtosis of the standard normal). Use this to prove that \(Var\left( Y \right)=2\) . Hint: \(Var\left( Y \right)=E\left( Y^2 \right)-E{\left( Y \right)}^2\) .

Solution

a. If \(Z ~ N\left( 0,1 \right)\) then \(E\left( Z \right)=0\) and \(Var\left( Z \right)=1\) . Since \(Var\left( Z \right)=E\left( Z^2 \right)-E{\left( Z \right)}^2\) we have

\[1=E\left( Z^2 \right)-0^2\]

or \(E\left( Z^2 \right)=1\) .

b. If \(Y ~ χ_1^2\) then \(Y\) is the square of a standard normal, that is, \(Y=Z^2\) and \(E\left( Y \right)=E\left( Z^2 \right)=1\) .

c. \(Var\left( Y \right)=E\left( Y^2 \right)-E{\left( Y \right)}^2\) . Since \(Y=Z^2\) , \(Y^2={\left( Z^2 \right)}^2=Z^4\) and

\[E\left( Y^2 \right)=E\left( Z^4 \right)=3\]

\(E{\left( Y \right)}^2=1^2=1\) and \(Var\left( Y \right)=3-1=2\) .