Optimal investment method: The following situation illustrates that different ways of averaging the same quantity can lead to significantly different results. Suppose one wishes to invest money by buying at the beginning of each month a certain number of shares of a company. The cost per share depends, of course, on the particular month and varies from month to month in a basically unpredictable way. There are two investment methods that we could think of: In method A, one buys the same number of shares each month; in method B, one buys shares for the same amount of money each month. After N months, one would have obtained a total of S shares and paid a total amount M of money. The best investment method is, naturally, the one that gives the greatest number of shares for the least amount of money; that is, the one that gives the largest ratio of S/M.

a) Obtain an expression for the ratio S/M in the case of method A. b) Obtain an expression for the S/M ratio in the case of method B. c) Show that method B is a better investment method, no matter how the cost fluctuates from month to month.

(We are kind of seeing basic probability for termophysics) Thank you!!

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