0

I want to maximize $\phi(c_{1},c_{2}) = \Phi(c_{1})\Phi(c_{2})$ given $c_{1}+c_{2} = 1$, where $\Phi(\cdot)$ is the CDF of standard normal distribution and $0\leq c_{i}\leq 1$ for $i=1,2$. I did it using simulations and found the solution at $c_{1} = c_{2}$. Is there any way to solve it analytically?

Satya Prakash
  • 367
  • I think that rewriting your expression as $\phi(c)$ = $\Phi(c)\Phi(1-c)$ would be a good place to start. Then probably differentiating. Interesting problem. – Leonhard Euler Jan 07 '18 at 14:46

1 Answers1

0

This is not rigorous at all, but I think the intuition is the following. When you differentiate and set equal to 0, you get $$ \frac{\phi(c)}{\Phi(c)} = \frac{\phi(1-c)}{\Phi(1-c)} $$ Then, seems there are two possibilities:

  • c = 1-c

  • $ \frac{\phi(x)}{\Phi(x)} = z $, with $z$ being some constant

Since the second one does not hold, seems like the first one is the right answer.

Paul
  • 177