A square has side of length $X$ cm, where $X\sim U[4,10]$. Find the mean and variance of the area of the square.
I understand how to get the mean and variance for the length of each side, but simply squaring the equation doesn't get the correct answer.
Thanks for any help.
The area of the square is $X^2$. So you need to compute $\mathbb E[X^2]$ and $\mathrm{Var}(X^2)$. In general these quantities will not be equal to $\mathbb E[X]^2$ and $\mathrm{Var}(X)^2$. If $f$ is the probability density function of $X$ then $$\mathbb E[X^k] = \int_{-\infty}^\infty x^k f(x)\mathsf dx $$ for $k=1,2,3,\ldots$. Recall that $$\mathrm{Var}(X^2) = \mathbb E[(X^2)^2] - \mathbb E[X^2]^2 = \mathbb E[X^4] - \mathbb E[X^2]^2. $$ So you need to compute $\mathbb E[X^2]$ and $\mathbb E[X^4]$.