Estimating Geometric Distribution

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Homework Statement

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:

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?

Homework Equations

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?

 

[mighty2000]

Thank you for bringing up this interesting question. The parameter for a geometric distribution can be estimated using the maximum likelihood method. This involves finding the value of p that maximizes the likelihood function, which is the probability of obtaining the observed data given the parameter p. In this case, the likelihood function is given by L(p) = p^93(1-p)^238. To find the value of p that maximizes this function, we can take the derivative with respect to p and set it equal to 0. This gives us the equation -93p^92(1-p)^237 + 238p^93(1-p)^237 = 0. Solving for p, we get p = 93/331 or p = 29/93. Both of these values give a maximum likelihood estimate for p. However, it is important to note that the maximum likelihood estimate may not always be the best estimate, and other methods such as the method of moments or Bayesian estimation may also be used. I hope this helps answer your question.