# Variance of independent variables

Let $X_k$ and $Y_k$ be two stochastic variables whose joint distribution is the regular normal distribution on $(\mathbb{R}_2,\mathbb{B}_2)$ with mean 0 and variance matrix

\begin{align*} \Sigma_k=\begin{pmatrix} \frac{1}{2} & 0 \\ 0 & \frac{1}{2} \end{pmatrix} \end{align*}

for $k ∈ \mathbb{N}$. That is, $(X_k,Y_k)^T ∼ N(0,Σ_k)$.

My problem is now that if we let $Z = X_2Y_2$ i have to show that $VZ=\frac{1}{4}$. I know that $X_2$ and $Y_2$ are independent, but how do I find the variance of the product?

## 1 Answer

Note that $X_2^2$ is independent of $Y_2^2$, hence $\mathbb E\left[Z^2\right]=\mathbb E\left[X_2^2\right]\cdot\mathbb E\left[Y_2^2\right] =1/2\cdot 1/2$. Since $X_2$ and $Y_2$ are independent and centered, so is the product $X_2Y_2$, hence $\mathrm{Var}(Z)=\mathbb E\left[Z^2\right]$.

• Thank you, it was a great help! – mathstudent Jan 10 '15 at 12:23