How to find MLE, probability distribution, and bias? Suppose the data consist of a single number $X$, from the following probability density:
$$f(x|θ) = \begin{cases}
\frac{1+xθ}{2} & & \text{for } -1 \leqslant x \leqslant 1, \\[6pt]
0 & & \text{otherwise},
\end{cases}$$
where $-1 \leqslant \theta \leqslant 1$.  Find the maximum likelihood estimate (MLE) $\hat{\theta}$ of the parameter $\theta$ and find its (exact) probability distribution.  Is the MLE unbiased?  Find its bias and MSE.  [Hint: First find the MLE for a few sample values of $X$; that should suggest to you the general solution.  Drawing a graph helps!  The distribution of the MLE will of course depend upon $\theta$.]
How do I begin?
 A: Begin with the hint given. Take $x=0.3$. Can you find the value of $\theta\in[-1,1]$ that maximizes $\frac{1+0.3\theta}{2}$? 
Take $x=0.8$. Can you find the value of $\theta\in[-1,1]$ that maximizes $\frac{1+0.8\theta}{2}$? 
Take $x=-0.5$. Can you find the value of $\theta\in[-1,1]$ that maximizes $\frac{1-0.5\theta}{2}$? 
Take $x=-0.9$. Can you find the value of $\theta\in[-1,1]$ that maximizes $\frac{1-0.9\theta}{2}$? 
A: The first step of finding the MLE is to write out the likelihood function (or log-likelihood function).  In the present problem, with only a single observed data point, the latter is given by:
$$\ell_x(\theta) = \ln(1 + x \theta) + \text{const.} \quad \quad \quad \text{for } -1 \leqslant \theta \leqslant 1.$$
This is the function you need to maximise to obtain the MLE.  Maximising this function is a univariate calculus problem, which will require you to derive the first and second derivatives of the function.  By maximising this function you will obtain the MLE as a function of the observed data value $x$, and you can then proceed to the next parts of the question.  (In this case, you will see that the maximising value is a "boundary solution" to the problem.)
