and then, come up with the fact that the log
likelihood from mu, disregard any terms that, that don't involve mu.
And I'm hoping everyone could come with the fact that the likeli,
the log likelihood for mu looks like this, bottom equation right here.
Okay.
And you know, you can, let's solve for a maximum likelihood estimate
so, the easiest way to do that right now would be to take the derivatives, set the
derivatives equal to zero and you get this answer.
X1 times r1 plus x2 times r2 divided by r1 plus r2, or in other words, x.
Times p plus x2 times 1 minus p, where p is r1
over r2 plus r2 and 1 minus p is r2 over r1
plus r2. And in this case, ri is 1 over sigma
squared sub i and then p is, of course, 1
over r1 divided by r1 plus r2. So, why does this makes sense?
This makes a lot of sense to me now.
but the first time you see it, you might say this makes
no sense but let me describe why this makes a ton of sense?
Okay, so notice what each ri is. It's 1 over the variance.
So,
if let's say sigma one
is huge. In other words that scale
stinks, it has this huge variants, then the weight r1
times x1. The weight given to the measurement from
that scale is very low. And then conversely you know, if, if sigma
2 is very small.
Then, I get r1 which is 1 over sigma squared will be a huge number
and then, x2 is given a gigantic weight and then we divide by r1 plus r2.
So that, so that when we weight these two things.
X1 and x2.
We, we get a convex combination, p times x1 plus 1 minus p times x2.
So, that it's an average.
It's just a weighted average.
Okay?