14. 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 rst ight and preceded by a ight.

The author then proposed and t a geometric probability distribution [p(x)  P(X  x)  px1 # q for x  1, 2, . . . , where q  1  p] to the data. The total sample size was n  130.

x 1 2 3 4 5 6 7 8 9 10 11 12 Number of Times x 48 31 20 9 6 5 4 2 1 1 2 1 Observed

a. The likelihood is 

. Show that the mle of p is given by

 , and compute for the given data.

b. Estimate the expected cell counts using of part

(a) [expected cell counts for x1, 2, . . . ], and test the t of the model using a x2 test by combining the counts for x7, 8, . . . , and 12 into one cell (x  7).