# Expected value from variance?

The variance for an unknown random variable $X$ is 6. The unknown random variable $X$ is for one side of a rectangle. The expected value for the area of the rectangle is 6 and the sum of all the sides is 20. What is the expected value of the unknown random variable $X$?

This problem seems simple, but I cannot wrap my head around it. I think I should use the algebraic formula for the variance $\mathrm{var}(X)=E(X^2)-E(X)^2$. The $\mathrm{var}(X)$ is known, but I can't figure out how implement the expected value for area in this. Any help?

This system of equations in $\mathbb{E}\left[X\right]$ and $\mathbb{E}\left[X^2\right]$ should help:

$$\mathbb{E}\left[X(10-X) \right] = 6$$ $$\mathbb{E}\left[X^2 \right] - \mathbb{E}\left[X\right]^2 = 6$$

Spoiler:

$$10\mathbb{E}\left[X \right] - \mathbb{E}\left[X^2 \right]= 6$$

• The first one gets me the expected value of area, that I know. How can I implement it to the second equation? I know I should get the answer from E(X)^2, but how do I implement the first equation to the E(X^2)? – Ryan Jan 29 '14 at 23:37
• Push the expectation through as much as you can in the first equation (x(10-x)=10x-x^2). – ir7 Jan 30 '14 at 0:09