Question

Need help with an statistical question!

Two dependable probability variables X and Y, who have the standard deviation of o-x=0.87 o-y=0.38 and the correlation coefficient equals 0.09. Calculate the standard deviation of X+Y.

The answer should be 0.9802

Answer 1

By "o-x" and "o-y" I assume you mean \(\sigma_X\) and \(\sigma_Y\), respectively, as in the standard deviations of \(X\) and \(Y\)?

Answer 2

yes excactly!

Answer 3

For two dependent random variables \(X\) and \(Y\), the variance of their sum is equal to the sum of their variances as well twice the covariance:
\[V(X+Y)=V(X)+V(Y)+2\text{ Cov}(X,Y)\]
where the covariance can be expressed in terms of the the given correlation coefficient:
\[\text{Cov}(X,Y)=\rho_{X,Y}\sigma_X\sigma_Y\]
(\(\rho_{X,Y}\) denotes the correlation coefficient.) On top of that, the standard deviation is equivalent to square root of the variance.

So, you have
\[\begin{align*}V(X+Y)&=V(X)+V(Y)+2\rho_{X,Y}\sigma_X\sigma_Y\\\\
{\sigma_{X+Y}}^2&={\sigma_X}^2+{\sigma_Y}^2+2\rho_{X,Y}\sigma_X\sigma_Y\\\\
\sigma_{X+Y}&=\sqrt{(0.87)^2+(0.38)^2+2(0.09)(0.87)(0.38)}\\\\
&=0.9802
\end{align*}\]

Answer 4

Thanks you so much for the explantation!