consider a mix [0, 1] of a random payoff X and a fixed amount c.

The payoff of a mix is

X = (1 )X + c.

Let f denote the probability density function of X.

where u(z) = 2z, c = 0.42, X is normal with mean 0.5 and variance 0.5. What is the optimal ?

U(f) = Eu(X) = integral u(z)f(z)dz.