Estimating Geometric Distribution

1. Nov 18, 2009

BlueScreenOD

1. The problem statement, all variables and given/known data
We return to the example concerning the number of menstrual cycles up to pregnancy, where the number of cycles was modeled by a geometric random variable. The original data concerned 100 smoking and 486 nonsmoking women. For 7 smokers and 12 nonsmokers, the exact number of cycles up to pregnancy was unknown. In the following tables we only incorporated the 93 smokers and 474 nonsmokers, for which the exact number of cycles was observed. Another analysis, based on the complete dataset, is done in Section 21.1.

Consider the dataset x1, x2, . . . , x93 corresponding to the smoking women,
where xi denotes the number of cycles for the ith smoking woman. The
data are summarized in the following table:
PHP:

Cycles    1   2   3   4   5   6   7   8   9   10  11  12
Frequency 29  16  17  4   3   9   4   5   1   1   1   3

The table lists the number of women that had to wait 1 cycle, 2 cycles,
etc. If we model the dataset as the realization of a random sample from a
geometric distribution with parameter p, then what would you choose as
an estimate for p?

2. Relevant equations

3. The attempt at a solution
The back of the text gives the following solution: One possibility is p = 93/331; another
is p = 29/93. But I'm not sure how this is derived, I found some very complicated formulas on Wikipedia for estimating the parameter of a geometric distribution, but they are beyond the scope of my textbook. Are there any basic formulas for computing the parameter for a geometric distribution?