Finding the Maximum Likelihood Estimate for Theta in a Random Sample of Size 8.

[semidevil]

I have a big test coming up, regarding estimators, but I just can't figure out the basics of maximum likilieehood.



so given this example, is this right?

p(k;theta) = theta^k * (1 - theta)^(1 - k), k = 0 1, and 0 < theta < 1.

so it's just the product of the function, and I get:

theta^K * 1 - (theta)^(sum from 1 to n of (k - n)).

then I take the natural log, and differentiate it to get

k/theta + (k - n)/ (1 - theta) = 0.

now, all I need to do is to put it in terms of theta...


first, did I do the first part right? in terms of finding the likelihoo function?

there's a lot of variables and I get confused when I do the products

 

[HallsofIvy]

It's hard to say- you haven't told us what n is or what k and theta are. Which is the parameter? Or are you doing a maximum likelihood estimate for both parameters at the same time?

In particular, I don't see any mention of a SAMPLE. The maximum likelyhood estimate for a parameter(s), given a SAMPLE, is the value of the parameter(s) that makes the probability of getting that particular sample largest.

 

[semidevil]

It's hard to say- you haven't told us what n is or what k and theta are. Which is the parameter? Or are you doing a maximum likelihood estimate for both parameters at the same time?

In particular, I don't see any mention of a SAMPLE. The maximum likelyhood estimate for a parameter(s), given a SAMPLE, is the value of the parameter(s) that makes the probability of getting that particular sample largest.

sorry...ok, so this is a random sample of size 8 w/ x1 = 1, x2 = 0, x3 = 1, x4 = 1, x5 = 0, x6 =1, x7 = 1, and x8 = 0.

I need to find the MLE for theta.