By David Machin (auth.)

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4. Expand (1 + 2/v)v as a binomial series and hence, by letting v be very large, show that lim v-+oo [1 + -vz)v 22 23 = 1 + 2 +-- +-- + ... 2! 3! What is the corresponding series for ~ vr 1 (1 + 2 5. A statistical distribution of great importance in many branches of biology, especially ecology, is the Poisson distribution whose successive terms are where r = 0, 1, 2, ••• oo and,\> 0. , r=O . 1 . ) ( Note: 6. Evaluate the terms of the binomial expansion of (1/2 + 1/2) 4 • Check that the terms sum to unity.

Calculate the population size after 5 generations on the following assumptions: (a) each mother cell continues to produce daughters; (b) the mother cell fails to produce daughters after the first division but nevertheless survives; (c) the mother cell produces daughters for two generations and then herself dies. (9) The total length (tip of snout to notch of flukes) and length from tip of snout to blowhole of seven adult male Humpback whales (Megaptera novaeangliae) are given in the following table.

7, which is taken from Bliss (1970, page 244). This shows that the mean of the square root of the number of normal human births in each hour of the day in a certain hospital is well fitted by a sine curve. 18) where t is the time in hours on a 24 hour clock (end of interval) and y is the mean square-root number of births. 18 is t (time) and not an angle. However, if we think of 24 hours as one complete cycle then 24 hours is equivalent to 2rr radians. Thus t = 0 corresponds to 8 = 0, t = 1 corresponds to 8 = 2rr/24 radians and so on.

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