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Let $X_1,X_2,X_3$ be uniform random variables on the interval $(0,1)$ with $$\newcommand{\cov}{\text{cov}} \newcommand{\var}{\text{var}}\cov(X_i,X_j)=\frac{1}{24} \text{ for } i,j\in\{1,2,3\}, i\ne j$$

Calculate the variance of $X_1+2X_2-X_3$

I did the below approach:

\begin{align*}\var(x_1+2x_2-x_3)&=\cov(x_1+2x_2-x_3,x_1+2x_2-x_3)\\&=\cov(x_1,x_1)+2\cov(x_1,x_2)-\cov(x_1,x_3)\\&+2\cov(x_2,x_1)+4\cov(x_2,x_2)-2\cov(x_2,x_3)\\&-\cov(x_3,x_1)-2\cov(x_3,x_2)+\cov(x_3,x_3)\\&= \frac{2}{24} -\frac{1}{24}+\frac{2}{24}-\frac{2}{24}-\frac{1}{24}-\frac{2}{24}=-\frac{2}{24}\end{align*} but the answer is $\dfrac{5}{12}$???

As $\cov(x_i,x_j)=\frac{1}{24} \text{ for } i\ne j$ so, $\cov(x_1,x_1), \cov(x_2,x_2),\cov(x_3,x_3)$ I assumed to be $0$. Is my assumption is wrong and I need to calculate the $\cov(x_1,x_1)$ from the UDF given.

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  • $\begingroup$ i kind of did that didn't i? $\endgroup$ – Prabir Acharya Jul 10 '16 at 6:53
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For any $X$, $\text{Cov}(X,X)=\text{Var}(X)$. So you for that part of the calculation you need to calculate the variance of the uniform distribution on $(0,1)$.

That variance is $\frac{1}{12}$, so you will be adding $\frac{1}{12}+\frac{4}{12}+\frac{1}{12}$ to the number you calculated. Note that $-\frac{2}{24}$ cannot be right: variance is always non-negative.

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  • $\begingroup$ ok!! so my assumption was wrong $\endgroup$ – Prabir Acharya Jul 10 '16 at 6:54
  • $\begingroup$ got the answer as 5/12, thanks for the help $\endgroup$ – Prabir Acharya Jul 10 '16 at 6:57
  • $\begingroup$ @PrabirAcharya: You are welcome. $\endgroup$ – André Nicolas Jul 10 '16 at 7:01

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