# Determine the correlation coefficient of the random variables X and Y if var(X) = 4, var(Y ) = 2, and var(X + 2Y ) = 15.

Determine the correlation coefficient of the random variables X and Y if var(X) = 4, var(Y ) = 2, and var(X + 2Y ) = 15.
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Nancy Phillips
The correlation coefficient of X and Y is
${\rho }_{X,Y}=\frac{Cov\left(X,Y\right)}{{\sigma }_{X}\ast {\sigma }_{Y}}$.
By the properties of variance
$Var\left(X+2Y\right)=Var\left(X\right)+{2}^{2}Var\left(Y\right)+2\ast 1\ast 2\ast Cov\left(X,Y\right)$
$={\sigma }_{X}^{2}+4{\sigma }_{Y}^{2}+4\rho {\sigma }_{X}{\sigma }_{Y}$,
and from the fact that
Var(X+2Y)=15
it follows that
${\sigma }_{X}^{2}+4{\sigma }_{Y}^{2}+4\rho {\sigma }_{X}{\sigma }_{Y}=15$
$4+4\ast 2+4\ast \rho \ast \sqrt{4}\sqrt{2}=15$
which implies that
$\rho =0.27$
Result:
$\rho =0.27$