# Covariance - Statistics

$X$ and $Y$ are random variables satisfying $Var (X) = 9, Var (Y ) = 25, Cov (X , Y ) = 10$. $U$ is defi ned as the average of $X$ and $Y$ , $U = \frac {1}{2} (X + Y )$.

Suppose a random variable $Z$ is de fined by $Z = aX + (1-a)Y$ . Calculate the value of $a$ which minimises $Var (Z)$.

The answer is 35/54 but I don't know how they arrived it.

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$\text{var}(Z) = a^2 \text{var}(X) + (1-a)^2\text{var}(Y) +2a(1-a)\text{cov}(X,Y)$ is a quadratic function of $a$. What value of $a$ minimizes it? –  Dilip Sarwate Feb 12 at 22:57