Find combination given expectation and variance

Question

I have found the expectation of X to be 4 and the Variance to be 3. For Y the expectation is 2 and variance is 2.

Is it possible to find a combination of X and Y which satisfies the expectation to be pi and variance to be sqrt(2)?

Answer 1

If by a combination you mean a variable like $aX+bY$ the answer is yes. Write the equations for the mean and variance of $aX+bY$ being equal to your values. You have two equations in two unknowns, $a,b$. Solve them.

Answer 2

Write down expressions for $E[aX + bY]$ and $\text{Var}(aX+bY)$ in terms of the expectation and variance of $X$ and $Y$. Then solve for $a$ and $b$.

$E[aX+bY] = aE[X] + bE[Y] = 4a+2b$ and $\text{Var}(aX+bY) = a^2 \text{Var}(X) + b^2 \text{Var}(Y) = 3a^2 + 2b^2$. If $4a+2b=\pi$ then $b=\pi/2-2a$. Then $3a^2+2b^2 = 3a^2 + 2(\pi/2-2a)^2$. Set this equal to $\sqrt{2}$ and solve for $a$.