Let $X_1,X_2$ be a random sample of size $n = 2$ from a distribution with density function given by:
$$f(x) = 2(x - q) , q \lt x \lt q + 1.$$
a) Show $E[(X - q)^k] = 2/(k+ 2)$ for $k \gt 0$.
b) Find the MME (method of moments estimate) for $q$.
c) Find the MLE (maximum likelihood estimate) for $q$. (The answer is not $(X_1 + X_2)/2$.)
The first part is no problem, and I get the sample mean $+ 2/3$ for the MME but struggling with part c) though. Any help is appreciated :).