Assume the marginal utility of $u(w)$ is $\alpha e^{-\alpha w}$, $\alpha > 0$ i.e. $u'(w)=\alpha e^{-\alpha w}$, $\alpha>0$.
a) Compute the utility function $u(w).$
b) Let $X_1$ and $X_2$ be two independent random variables following the normal distribution
$$X_1 \sim N(\mu_1,\sigma_1^2), X_2 \sim N(\mu_2,\sigma_2^2),$$ such that $\sigma_1=2$ and $\sigma_2=6$.
Compute the relation between $\mu_1 \ge 0$ and $\mu_2 \ge 0$ knowing that the premium for $X_2$ is twice the premium for $X_1$ and $\alpha =0.1$.
c) Under the same assumptions of the previous parts, assume that $\mu_1=1$ and $\mu_2=0,6$. Is it possible to determine for which values of $\alpha$ the premium $\prod_{X_1}$ is higher than $\prod_{X_2}$? If yes, find the values for $\alpha$, if it is not possible, justify why.
My Understanding
For (a) since it's supposed to be a utility function we know that $u(0)=0$, so one has $u(w)=1-e^{-\alpha w}$, $\alpha>0$.
Now for (b), I thought that I had to apply the zero utility principle, but I am not sure.