Constructing the likelihood function and find the MLE

Question

I am Matthew and I have to solve an exercise for my Econometrics2 course.

I am having a hard time finding the likelihood function and then deriving the maximum likelihood estimator of a sample. The information given in the question is somewhat confusing to me, hence I am not sure if my approach is correct.

The question is the following:

*Let Yi , i = 1, ..., n denote an i.i.d. sample and suppose that Yi = 1 with probability p and Yi = 2 with probability 1 − p. You want to estimate p.

1. Construct the likelihood function for p.
2. Derive the maximum likelihood estimator of p.*

Thanks for your suggestions in advance!

Answer 1

First Method

Your density is the following

$$\mathbb{P}[Y=y]=p^{2-y}(1-p)^{y-1}$$

$y=\{1;2\}$

Your likelihood is

$$L(p)=p^{2n-\Sigma_i Y_i}(1-p)^{\Sigma_i Y_i-n}$$

take its log, derivate w.r.t. $p$, set =0 and solve in $p$ obtaining

$$\hat{p}_{ML}=2-\overline{Y}_n$$

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Second Method

You can transform you rv in the following

$$Y-1 =Z= \begin{cases} p, & \text{if $z=0$ } \\ 1-p, & \text{if $z=1$ } \end{cases}$$

Now your Z is a bernulli with parameter $\pi=(1-p)$

It is well known that

$$\hat{\pi}_{ML}=\frac{\Sigma_i Z_i}{n}=\overline{Y}_n-1$$

Now using invariance property you get immediately

$$\hat{p}_{ML}=1-\hat{\pi}_{ML}=1-\overline{Y}_n+1=2-\overline{Y}_n$$

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As you can see, the two results are the same.