Suppose that $X$ has conditional cdf $$ F(x \mid \theta) \propto 1-e^{-\theta \sin x}, 0 \leq x \leq \pi / 2 ; \theta>0 $$ (a) Find the conditional pdf $f(x \mid \theta)$ of $X$.
(b) Given $x_1, x_2, \cdots, x_n$, a random sample of observation of $X$, derive an equation which the MLE $\hat{\theta}$ of $\theta$ must satisfy.
(c) Find an approximate expression for the posterior variance of $\theta$ for the case where $n$ is large. You may assume that the prior density of $\theta$ is non-zero for $\theta>0$.
For part (a), I solve it by taking derivatives of CDF, and get $f(x|\theta) = \theta \cos x e^{-\theta \sin x}$
For part (b), I got $\hat{\theta} = \frac{n}{\sum \sin x_i}$.
But for part (c), I don't know how to derive it without a specific prior distribution. Maybe the part is asking for Cramer-Rao Lower Bound, but I am not sure about it.