1. ## question about likelihood function/MLE

If a random variable Y has the following probability distribution:
P (Y = 1) = p, P (Y = 2) = q, and P (Y = 3) = 1−p−q. A random sample of size
n is drawn from this distribution and the random variables are denoted Y1, Y2, · · · , Yn.

How would you derive the likelihood function for p and q? Would you just sum the probability distributions first?

How would you derive the MLE for p and q?

If someone could get me in the right direction that would be appreciated

2. Ok. So now you are given

The probability mass function of is

However, you can write a more condensed product form:

such that the joint log-likelihood for n sample

where

Now we take the partial differentiation with respect to the parameters
and set them equal to 0:

Simplify them, we have a 2 by 2 linear system in :

Eventually, we have

Again note

But note that a, b, c are just counts of the outcomes 1, 2, 3.
So the MLE are just the ratio of the specific counts to the total sample.

3. how did you derive the condensed product form?

4. Well just look what happens to it when you plug in the values y=1, y=2, y=3. You get what you want. Notice that for y = 1 we want it to be p. If we plug in y = 2 we get p^{(2-2)(2-3)/2} = p^0, if we plug in y = 3 we get p^0 again. It's just being clever about how you set up the exponents so that things cancel and give you what you want.

5. if someone could verify my answer to the MLE for p and q that would be much appreciated,

p = (1-q)/2

q = (1-p)/2

6. I guess the most important thing here is not the tricky product form.
It just ease my presentation.

In fact, without mentioning that we can also write down the likelihood
function, which is the most important thing you need to know:
the exponents of each outcome in those kind of "multinomial" distribution
are just the observed frequency for each outcome.

PS: Sorry not quite understand the follow up post.
Anyway it is an easy linear system and you can solve it to get

7. Originally Posted by messianic
if someone could verify my answer to the MLE for p and q that would be much appreciated,

p = (1-q)/2

q = (1-p)/2
Those don't really look like estimators to me. Since you don't know either p or q to begin with how would you calculate these once you gather data?

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