# Determine variance & mean

I am not sure how to determine the variance and mean for the following equation system. I'd greatly appreciate your help!

$f_\xi(x) = \left\{ \begin{array}{l l} 10k_1x & \quad \text{for$0<x<1$}\\ 0 & \quad \text{for other}\\ \end{array} \right.$

$f_\eta(x) = \left\{ \begin{array}{l l} 4k_2x & \quad \text{for$0<x<1$}\\ 4k_2(2-x) & \quad \text{for$1\leq x<2$}\\ \end{array} \right.$

I have determined the following:

$k_1 = \frac{1}{5}$

$k_2=\frac{2}{7}$

... and I believe to have found their individual variance and mean:

$E_\xi(x)=\frac{2}{3}$

$V_\xi(x)=\frac{1}{18}$

$E_\eta(x)=\frac{22}{21}$

$V_\eta(x)=\frac{145}{882}$

Now to the gist of my question: how do I calculate the variance and the mean for $10\xi+2\eta$? Thank you!

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Since it is homework, I will not calculate mean, variance apart from giving the odd hint. But for the second, it should be obvious by symmetry that the mean is $1$. If you did it by integration, there was a slip. For the variance, often the easiest thing is to use $E(X^2)-(E(X))^2$. –  André Nicolas May 20 '12 at 18:15

## 1 Answer

Let $X$ and $Y$ be random variables, and let $a$ and $b$ be constants.

We have $E(aX+bY)=aE(X)+bE(Y)$.

If $X$ and $Y$ are independent then $\text{Var}(aX+bY)=a^2\text{Var}(X)+b^2\text{Var}(Y)$.

(If $X$ and $Y$ are not independent, one can say little about the variance of the linear combination.)

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Not so Andre Var(X+Y)=Var(X)+Var(Y)+2 Cov(X,Y) and so Var(aX+bY)=a^2 Var(X) + b^2 Var(Y) + 2ab Cov(X,Y). Of course he needs to calculate the covariance in his example. –  Michael Chernick May 20 '12 at 16:30
@MichaelChernick: Agreed! However, the density function information provided in the problem tells us nothing much about the covariance, apart from the very weak bounds of Cauchy-Schwartz. –  André Nicolas May 20 '12 at 18:12
You are right I thought he specified the joint distribution and not just the two marginals. –  Michael Chernick May 20 '12 at 19:40