medium Probability
A release-checked medium problem for training Continuous Random Variables, Expected Value.
Suppose that you generate a uniformly random number in the interval $(0,1)$. You can generate additional random numbers as many times as you want for a fee of $\$0.05$ per generation. The number of additional generations must be made prior to your first number that you generate. Your payout is the maximum of all the numbers you generate. Assuming optimal strategy, what is your expected payout of this game?
This Probability problem is tagged Continuous Random Variables, Expected Value. State the random variables and conditioning information explicitly, then check the result against boundary cases before opening hints or a solution.
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