Maximum likelihood estimate help

Carl R. · science forum beginner Joined: 01 May 2005 Posts: 38

Hello, I'm stuck with the following problem , I don't know if my answer
is correct, could you please help me?

An urn contains black and white balls. A sample of size n is drawn with
replacement.
What is the maximum likelihood estimator of the ratio R of black to
white balls in the urn?

So I think the distribution is given by C(n,x) * p^x* (1-p)^(n-x) where
p is the probability of getting a black ball.

For the maximum likelihood estimator of the ratio R of black to white
balls in the urn I get p/(1-p) where p= (number of black balls)/n. Is
this correct?

Then it says:

Suppose that one draws balls one by one with replacement until a black
ball appears. Let X be the number of draws required (not counting the
last draw). This operation is repeated n times to obtain a sample X_1,
X_2,..,X_n. What is the maximum-likelihood estimator of R on the basis
of this sample?

I think the distribution of getting a black ball is given by
C(n,x) * (w/(w+b))^x * (1 - w/(w+b))^(n-x) / (w+b)^n

where w= number black balls, b= number white walls.

Then I get X/ (n-X) for this one. Is this incorrect?

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