1) Example: The Normal Distribution. Last week you calculated the mean and variance of the chest size of Scottish Militia men (Quetelet, et al., 1846). The probability distribution was discrete; chest size was measured to the nearest inch. You found that the mean of this distribution is 39.83 and its variance is 4.20 (standard deviation 2.05). The distribution is remarkably bell-shaped. Assume that chest size, when measured as a continuous variable in this population, is normally distributed with mean and variance given above.
a) Normal Probabilities. If the militia does not stock jackets for soldiers with chest sizes smaller than 35.5 and larger than 43.5 inches, what is the probability that it will not have a jacket for a randomly arrived new recruit? You can use the JAVA applet written by Gary McClelland of The University of Colorado, Boulder to answer this question.
b) Normal Quantiles. If the militia wants to be 99% sure they have in stock a jacket large enough for a randomly arriving new recruit, how large a chest size must their jackets accommodate? If they want to be 95% sure they stock a jacket small enough for a random new recruit, how small a chest size must their jackets accommodate? You can use the Quantile JAVA applet, written by Balasubramanian Narasimhan of Stanford University, to answer this question.

2) Example: The Binomial Distribution. Suppose that, in a particular age and gender subgroup of the population, annual mortality is 1%. Suppose a 1 year term life insurance policy with a $5100.00 benefit is priced at $100.00 for individuals in this population group (individuals pay $100 at year's beginning and if they do not survive the year their beneficiaries are payed $5100.00; the same example given in class). A company sells a number, n, of these policies. Define X to be the number of policies that require payment at year's end. Neglecting various costs, profit on n policies, P = (100*n) - (5100*X). X is a random variable with the binomial distribution (we are looking at the example from class in a different way, one that makes it easier to calculate the probability of a loss). What is its "success" probability? Suppose that there are two firms: Firm A sells 10 such policies and Firm B sells 50.
a) What are expected profits for the two firms?
b) Use the binomial probability JAVA applet written by Balasubramanian Narasimhan of Stanford University to answer: approximately what is the the probability that Firm A will make money on its 10 policies; what is the probability that Firm B will make money on its 50 policies? Use formula 4-8, in the book to answer this question exactly.