# self learning – MAP and MMSE Estimations

Let the observation $$x$$ has the following density:
$$begin{equation} p(x|theta)= begin{cases} frac{1}{theta},& text{if } 0 < x leq theta\ 0, & text{otherwise} end{cases} end{equation}$$
and the r.v $$theta$$ has:
$$begin{equation} p(theta)= begin{cases} theta exp(-theta),& text{if } 0 leq theta\ 0, & text{otherwise} end{cases} end{equation}$$
I try to find MAP and MMSE estimation of the parameter $$theta$$.

MAP:

$$hat{theta} = argmax_{theta}$$ $$p(theta|x) = argmax_{theta}$$ $$p(x|theta)p(theta) = argmax_{theta}$$ $$exp(-theta)$$ which gives $$hat{theta} = 0$$. But it seems $$p(x|theta)$$ is not well defined at $$theta = 0$$. Am I missing something here?

MMSE:

$$hat{theta} = E(theta | x) = int_{0}^{infty} theta exp(-theta) ,dtheta = 1$$. I want to verify if it’s correct.