Problem source

An industrial process produces rectangular plates of mean length $\mu_{1}$ and mean breadth $\mu_{2}$. The length and breadth vary independently with non-zero standard deviations $\sigma_{1}$ and $\sigma_{2}$ respectively. Find the means and standard deviations of the perimeter and of the area of the plates. Show that the perimeter and area are not independent.

Solution source

Let $L \sim N(\mu_1, \sigma_1^2)$, $B \sim N(\mu_2, \sigma_2)^2$, so

\begin{align*}
&& \mathbb{E}(\text{perimeter}) &= \E(2(L+B)) \\
&&&= 2\E[L]+2\E[B] \\
&&&= 2(\mu_1+\mu_2) \\
&&\var[\text{perimeter}] &= \E\left [ (2(L+B))^2 \right] - \left ( \E[2(L+B)] \right)^2 \\
&&&= 4\E[L^2+2LB+B^2] - 4(\mu_1+\mu_2)^2 \\
&&&= 4(\sigma_1^2+\mu_1^2+2\mu_1\mu_2+\sigma_2^2+\mu_2^2) - 4(\mu_1+\mu_2)^2\\
&&&= 4(\sigma_1^2+\sigma_2^2) \\
&&\text{sd}[\text{perimeter}] &= 2\sqrt{\sigma_1^2+\sigma_2^2} \\
\\
&& \E[\text{area}] &= \E[LB] \\
&&&= \E[L]\E[B] \\
&&&= \mu_1\mu_2 \\
&& \var[\text{area}] &= \E[(LB)^2] - \left (\E[LB] \right)^2 \\
&&&= \E[L^2]\E[B^2]-\mu_1^2\mu_2^2 \\
&&&= (\mu_1^2+\sigma_1^2)(\mu_2^2+\sigma_2^2) -\mu_1^2\mu_2^2 \\
&&&= \sigma_1^2\mu_2^2 + \sigma_2^2\mu_1^2 + \sigma_1^2\sigma_2^2\\
&& \text{sd}(\text{area}) &= \sqrt{\sigma_1^2\mu_2^2 + \sigma_2^2\mu_1^2 + \sigma_1^2\sigma_2^2} \\
\\
&& \E[\text{perimeter} \cdot \text{area}] &= \E[2(L+B)LB] \\
&&&= 2\E[L^2]\E[B] + 2\E[L]\E[B^2] \\
&&&= 2(\sigma_1^2+\mu_1^2)\mu_2 + 2(\sigma_2^2+\mu_2^2)\mu_1 \\
&& \E[\text{perimeter}] \E[\text{area}] &= 2(\mu_1+\mu_2) \cdot \mu_1\mu_2
\end{align*}

Since the latter does not depend on $\sigma_i$ but the former does they cannot be equal in general, therefore they cannot be independent.

[See also STEP 2006 Paper 3 Q14]