Maximum Likelihood Estimate of a a discrete r.d - I spent more than 4 hours on this questions, help!!

Question

Suppose $X$ is a discrete r.d with the following p.d.f:

$$ \begin{array}{c|lc} X & \text{0} & \text{1} & \text{2} & \text{3} \\ \hline p(x) & 2\theta/2 & \theta/3 & 2(1-\theta)/3 & (1-\theta)/3 \end{array} $$

where $0\leq\theta\leq1$ was taken from such a distribution $(3,0,2,1,3,2,1,0,2,1)$. What is the MLE of $\theta$?

Likelihood function: $$(2\theta/2)^2.(\theta/3)^3.(2(1-\theta)/3)^3.((1-\theta)/3)^2$$ Log-Likelihood function: $$2\log(2\theta/2)+3\log(\theta/3)+3\log(2(1-\theta)/3)+2\log((1-\theta)/3)$$ Set derivative to zero: $$2/\theta+3/\theta-6/(2-2\theta)-2/(1-\theta)=0$$ $$\frac{5-5\theta}{\theta}-\frac{6+6\theta}{2-2\theta}=2$$ $$5-5\theta=2\theta+\frac{6\theta+6\theta^2}{2-2\theta}$$ $$2-2\theta(5-7\theta)=6\theta+6\theta^2$$ Finally I got something like this: $$10+8\theta^2=30\theta$$ Don't know how to find $\theta$. But I guess I have made mistake because the answer is $\theta=0.5$.

I spent more than 4 hours on this, tried different ways, like simplify the likelihood function before $"\log"$ them and still cannot work out, anyone help please?

Answer 1

Your calculation could be greatly simplified if you first algebraically simplify the likelihood while ignoring all constants of proportionality (i.e. factors of the likelihood that are constant with respect to the parameter). That is to say, we can write $$\mathcal L(\theta \mid \boldsymbol x) \propto \theta^5 (1-\theta)^5,$$ from which we obtain $$\ell(\theta \mid \boldsymbol x) = 5 \left( \log \theta + \log (1-\theta) \right).$$ Now it is trivial to compute $$\frac{\partial \ell}{\partial \theta} = 5 \left( \frac{1}{\theta} - \frac{1}{1-\theta} \right) = 0,$$ hence $\hat \theta = 1/2$ as claimed.

Note also that, although it makes no difference to the likelihood whether $\Pr[X = 0 \mid \theta] = 2\theta/2$ or $2\theta/3$, it is important to understand in a qualitative sense why it makes no difference. For had it been $\theta + c$ for some constant $c$, then it does affect your MLE estimation. Why? The reason is that when $\theta$ is regarded as a variable quantity (as it is in the likelihood, given the observed sample), we see that the relative likelihood is unaffected by any scaling factors.

This of course does not excuse the error of the sum of probabilities not adding to unity.

Answer 2

Your $p(x)$ is wrong: its sum is $4/3$, not $1$. And who would write $2\theta/2$ instead of $\theta$? I suspect that $2\theta/2$ should be $2\theta/3$ and $2(1-\theta)/2$ should be $2(1-\theta)/3$.