The question is: Let k be an unspecifed constant, and suppose we are given that A;B;C are three random variables that satisfy C = (kA + 3B)2;E[A] = E[B] = 0; var(A) = 4; var(C) = 16; correlation(A;B) = -0.5: Determine the value of k that minimizes E[C].
I have determined that E[A]2 =4 and E[C] = 4k^2 +6k(E[AB])+ 9(E[B^2]). Is anyone able to help me how to find out what is the value of E[AB] and E[B^2]? Thank you.
You have $-0.5 = cor(A, B) = \frac{cov(A, B)}{\sqrt{var[A]var[B]}} = \frac{E[AB] - E[A]E[B]}{\sqrt{var[A]var[B]}} = \frac{E[AB]}{\sqrt{4var[B]}} \Rightarrow E[AB] = -\sqrt{var[B]}$.
Now, you need to optimize a function (which is a convex quadratic function, so the minima exists)
$$ E[C] = 4k^2 + 6k(-\sqrt{var[B]}) + 9var[B], $$
where $k$ is the function argument, and $var[B]$ is an unknown constant and is treated as a parameter. Then,
$$ \frac{d}{dk}E[C] = \frac{d}{dk}\left(4k^2 + 6k(-\sqrt{var[B]}) + 9var[B]\right) = 8k - 6\sqrt{var[B]} = 0 \Leftrightarrow k = \frac{3}{4}\sqrt{var[B]}. $$
$k$ depends on the $var[B]$.