The article "Feeding Ecology of the Red-Eyed Vireo and Associated Foliage-Gleaning Birds" (Ecological Monographs, 1971: 129-152) presents the accompanying data on the variable X = the number of hops before the first flight and preceded by a flight. The author then proposed and fit a geometric probability distribution [p(x) = P(X = x) = px-1 €¢ q for x = 1, 2,..., where q = 1 - p] to the data. The total sample size was n = 130.

a. The likelihood is (px1-1 €¢ q)? ... ?(pxn-1 €¢ q) = pˆ‘xi-n €¢ qn. Show that the mle of p is given by (ˆ‘xi - n) / ˆ‘xi, and compute for the given data.
b. Estimate the expected cell counts using of part (a) [ expected cell counts = n . ()x-1 . for x = 1,2,...], and test the fit of the model using a x2 test by combining the counts for x 5 7, 8,..., and 12 into one cell (x ‰¥ 7).

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1 2 3 4 56789 10 11 12 Number of Times r 48 31 20 9 6 5 421 1 2 1 Observed