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likelihood using data

Hi

The probabilities of sets of twins are as follows.

Both Male: 1/4(1+theta)
Both Female: 1/4(1+theta)
Mixed Sex: 1/2(1-theta)

From a study of 200 pairs if twins:

Both Male= 78
Both Female= 59
Mixed= 63

Find the Likelihood and Maximum Likelikodd estimator for the data.

I've never dealt find findng likelihoods of datas, just of functions.
Can anyone help?!
I've never done anything like this myself as I don't do Stats any more, but I would suggest that a good starting point would be to find theta?
Reply 2
Avatar for choccookies
choccookies
OP
yeah but how do i go about that?
You've been given the probabilities of every possible scenario in terms of theta, so their sum must be 1.
Reply 4
Avatar for choccookies
choccookies
OP
i've tried that but it came to 0, so im guesing im doing something completely wrong;
1=1/4(1+theta) + 1/4(1+theta) +1/2(1-theta)
0=theta/4 - theta/2
0=0

:s
Actually it would give you theta/2 = theta/2, but oh well. Sorry I couldn't help. :colondollar:
Reply 6
Avatar for SsEe
SsEe
13
You have a random variable X which can take 3 values (both male, both female or mixed) with the probabilities given.
Then you have a set of observations, x1, x2, x3, ..., x200. So x1,..., x78 are "both male" etc
L(theta) = (Product over the xi) P(X=xi)
Reply 7
Avatar for astudent
astudent
7
You have three separate events, which have probabilities in terms of a parameter, can you think how to combine the whole samplespace into the likelyhood function? Check out below for the likelyhood function if you cant see it [ Using x instead of theta].

Spoiler

Reply 8
Avatar for choccookies
choccookies
OP
SsEe
You have a random variable X which can take 3 values (both male, both female or mixed) with the probabilities given.
Then you have a set of observations, x1, x2, x3, ..., x200. So x1,..., x78 are "both male" etc
L(theta) = (Product over the xi) P(X=xi)


I dont understand how to write this out.
P(X=78)= 1/4(1+theta)
P(X=59)= 1/4(1+theta)
P(X=63) = 1/2(1-theta)

So in the likelihood equation I do product of these multiplied together? That obviously isnt right because then I would have no x to deal with .

Argh I'm so cinfused, and im sure its pretty straight forward!
Reply 9
Avatar for SsEe
SsEe
13
choccookies
I dont understand how to write this out.
P(X=78)= 1/4(1+theta)
P(X=59)= 1/4(1+theta)
P(X=63) = 1/2(1-theta)

So in the likelihood equation I do product of these multiplied together? That obviously isnt right because then I would have no x to deal with .

Argh I'm so cinfused, and im sure its pretty straight forward!


X can't be 78 or 59 or 63. X can be "both male", "both female" or "mixed". Call them 1, 2, 3 if you like. Then
P(X=1) = 1/4(1+theta)
P(X=2) = 1/4(1+theta)
P(X=3) = 1/2(1-theta)

The sample is just a load of values x1, x2, ..., x200 taking values 1, 2 or 3.
Eg, P(X=x1) = P(X=1) = 1/4(1+theta). P(X=x200) = P(X=3) = 1/2(1-theta)
So can you see what (Product over the xi) P(X=xi) is?

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