** Please see the attachment for the full problem description **
Consider the following game. A fair coin is ﬂipped until the ﬁrst tail appears; we win $2 if it appears on the ﬁrst toss, $4 if it appears on the second toss, and, in general, $2^k if it ﬁrst occurs on the kth toss Let the random variable X denote our winnings. How much should we have to pay in order for this to be a fair game? [Note: A fair
game is one where the difference between the ante and E (X ) is 0.] Known as the St. Petersburg paradox, this problem has a rather unusual answer.
First, note that
pX(2*)=P(X=2k)=2Lk, k= 1,2,...
(please see the attached file)
3.5.20. For the St. Petersburg problem (Example 3.5.5), ﬁnd the expected payoff if
(a) the amounts won are c^k instead of 2^k, where 0 < c < 2.
(b) the amounts won are log 2^k. [This was a modi-
ﬁcation suggested by D. Bernoulli (a nephew of
James Bernoulli) to take into account the decreasing
marginal utility of money—the more you have, the
less useful a bit more is.]
Note that in both parts, the probability function p(x) = 1/2^k does not change. We simply replace the winnings amount in the calculation of the expectation.
The summation becomes
E(X) = sum c^k * p(x) = sum (c/2)^k = c/2 + (c/2)^2 + (c/2)^3 + ...
This is an infinite geometric series with common ratio c/2. Read Wikipedia's link below to find out more about geometric series in ...
This solution solves the given probability problem.