Let R be a random variable giving the per capita production in a population with p.d.f. g(x) = 1.25 for 0.7 ‰¤ x ‰¤ 1.5 (the values used in Example 6.1.3). Are the simulations in Example 6.1.3 within this range?
As in Exercises 37 and 38, the Central Limit Theorem for sums can be used to approximate the logarithm of a product even when the random variables multiplied together are continuous random variables. Find the normal distribution that approximates the logarithm of the population size P50 assuming that P0 = 1. You will need the indefinite integrals

to evaluate the expectation and variance of ln(R). Use the rule of thumb that most populations end up within two standard deviations from the mean to give a range of probable population sizes.

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In(x)dx = x ln(x)-x , In(x)dx = x ln(x)--2x ln(x) + 2x