Thread: Maximum likelihood estimator of the ratio

Please help me is this statistical problem.

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?
Suppose that one draws balls one by one with replacement until a black appears.
Let X be the number of draws required (not counting the last draw).
This operation is repeated n times to obtain a sample X1, X2, ..., Xn.
What is the maximum-likelihood estimator of R on the basis of this sample.

Originally Posted by imjae

Please help me is this statistical problem.

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?
Suppose that one draws balls one by one with replacement until a black appears.
Let X be the number of draws required (not counting the last draw).
This operation is repeated n times to obtain a sample X1, X2, ..., Xn.
What is the maximum-likelihood estimator of R on the basis of this sample.

The s are independent identically distributed RVs with:



where

So now write out the likelihood of outcome ..

Originally Posted by CaptainBlack

The s are independent identically distributed RVs with:



where

So now write out the likelihood of outcome ..

Thank you for your reply.
May I know why p = R/(1+R)?
Can we let R be equal to b/w, where b=no. of black balls and w=no. of white balls?