A lake contains fishes of n different species. A fisheries biologist catches fishes from the lake...

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A lake contains fishes of n different species. A fisheries biologist catches fishes from the lake until she has caught one example from each species. Assuming that the probability of being caught is the same for all species, determine the expected number of fishes that need to be caught in total so that each species is represented with at least one example. For simplicity, we also assume that that the population of fishes is 'infinitely large' so that catching a new fish of a random species can be treated as independent from all previous catches. Hints. " Define the random variable X: 'number of fishes caught until all species are rep- resented with at least one example'. Show that X = Σ Zi with Zi ~ Geo(1 - (i-1)/n), i = 1,...,n, where the random variable Z represents the number of fishes caught between the 'discovery' of the i-th and i-1-th different species, i = 1, 2,...,..., n. ■ Use known results about the expectations of geometric random variables and lin- earity of expecation. A lake contains fishes of n different species. A fisheries biologist catches fishes from the lake until she has caught one example from each species. Assuming that the probability of being caught is the same for all species, determine the expected number of fishes that need to be caught in total so that each species is represented with at least one example. For simplicity, we also assume that that the population of fishes is 'infinitely large' so that catching a new fish of a random species can be treated as independent from all previous catches. Hints. " Define the random variable X: 'number of fishes caught until all species are rep- resented with at least one example'. Show that X = Σ Zi with Zi ~ Geo(1 - (i-1)/n), i = 1,...,n, where the random variable Z represents the number of fishes caught between the 'discovery' of the i-th and i-1-th different species, i = 1, 2,...,..., n. ■ Use known results about the expectations of geometric random variables and lin- earity of expecation.

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