 Solution

ID: past-exam-of-the-probability-statistics-and-options-i-course-of-the-university-of-oxford/2009/o/2/i/solution

With immediate replacement and uniform mixing, each capture is a Bernoulli trial with tagged probability

p = k / N

. Counting failures before the

r

th tagged capture gives the stated negative binomial distribution. Independent runs on the successive days give the sample model.

S = \sum_{i} x_{i}

, the log-likelihood is

ℓ (p) = n r lo g p + S lo g (1 - p) + constant

. Its derivative vanishes at

p = \frac{n r}{n r + S} = \frac{r}{r + x} .

The second derivative is negative. If

S = 0

, this formula gives the boundary maximum

p = 1

p < 1

, it is a supremum rather than an attained interior maximum.

Using the supplied negative-binomial mean,

E S = n r (1 - p) / p

. Thus total Fisher information for the sample is

I_{n} (p) = - E ℓ^{''} (p) = \frac{n r}{p ^{2}} + \frac{n r}{p ( 1 - p )} = \frac{n r}{p ^{2} ( 1 - p )} .

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