Let $X_1,X_2, \dots ,X_n$ be a random sample from the pdf
$$f(x,\gamma)=\frac{2\gamma+1}{2\gamma}x^{\frac{1}{2\gamma}},\quad 0\leq x \leq 1,\gamma>0,$$ what are the steps to find the Maximum Likelihood Estimator of the probability $P(X\leq 0.5)$?
All I can think of is integrating the pdf from $0$ to $0.5$, but then there'll be no $x_i$ and the likelihood function would be
$$L(\gamma)=n\int_0^{0.5}f(x,\gamma)\,dx$$
Is the solution of the equation
$$\frac{dL(\gamma)}{d\gamma}=0$$
the final answer?