# 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 $\displaystyle X_i$ s are independent identically distributed RVs with:

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

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

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

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

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

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

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