Problem 147 Solution

Lets rephrase the question to a more general case, what is the expected number needed for the sum to exceed x, where x<=1?

Obviously at least one number will be needed. If the first number is less than x then another will will need to be drawn with a lesser total needed to finish.

So E(x) = 1 + integral from 0 to x of E(y) dy.

Take the derivative of both sides...

E'(x) = E(x)

What function is a derivative of itself? e^x of course! So the answer is e¹ = e.

Thanks to Extra Stuff: Gambling Rambling by Peter Griffin for this problem. See chapter 11.

Michael Shackleford, ASA, August 20 1999

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