Determine the variance of the sum of two random variables, uniformly distributed in the range from $3$ to $9$.

Question

Determine the variance of the sum of two random independent variables, uniformly distributed in the range from $3$ to $9$.

I have no ideas, can you help me?

Answer 1

Let $X$ and $Y$ be independent $\mathrm{Unif}(3,9)$ random variables and denote $Z=X+Y$. Then $$ \mathbb E[X] = \frac{9+3}2 = 6, $$ $$ \mathbb E[X^2] = \int_3^9 \frac16x^2\ \mathsf dx = \frac1{18}(729-27) = 39, $$ so $$\operatorname{Var}(X) = \mathbb E[X^2] - \mathbb E[X]^2 = 39 - 6^2 = 3.$$ Since the variance of the sum of uncorrelated random variables is the sum of their variances, we have $$ \operatorname{Var}(X+Y) = \operatorname{Var}(X) + \operatorname{Var}(Y) = 3+3 = 6. $$