# Thread: Maximum likelihood estimator of the ratio

1. ## Maximum likelihood estimator of the ratio

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.

2. Originally Posted by imjae

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 $X_i$ s are independent identically distributed RVs with:

$p(X_i=k_i)=(1-\rho)^{k_i-1}\rho$

where $\rho=R/(1+R)$

So now write out the likelihood of outcome $k-1, k_2, .., k_n$ ..

3. Originally Posted by CaptainBlack
The $X_i$ s are independent identically distributed RVs with:

$p(X_i=k_i)=(1-\rho)^{k_i-1}\rho$

where $\rho=R/(1+R)$

So now write out the likelihood of outcome $k-1, k_2, .., k_n$ ..

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?

### an urn contains black and white balls. a sample of size n is drawn with replacement. what is the maximum likehood estimator of the ratio of black to white balls??

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