Statistics: Maximising expected value of a function of a random variable An agent wishes to solve his optimisation problem: $ \mbox{max}_{\theta} \ \ \mathbb{E}U(\theta S_1 + (w - \theta) + Y)$, where $S_1$ is a random variable, $Y$ a contingent claim and $U(x) = x - \frac{1}{2}\epsilon x^2$.
My problem is - how to I 'get rid' of '$\mathbb{E}$', to get something I can work with? Thanks

Expanding the comment by Ilya:
$$\mathbb{E}\,U(\theta S_1 + (w - \theta) + Y)
=\mathbb{E} (\theta S_1 + (w - \theta) + Y) - \frac{\epsilon}{2} \mathbb{E} \left((\theta S_1 + (w - \theta) + Y)^2\right)
$$
is a quadratic polynomial in $\theta $ with negative leading coefficients. Its unique point of maximum is found by setting the derivative to $0$.
