Suppose I play a game in which I flip a fair coin till I get tails. Let [MATH]N[/MATH] be the number of heads in the sequence of tosses. If I get a payoff of [MATH]\min(2^N, X)[/MATH], where [MATH]X[/MATH] is a constant like, say, [MATH]\$500[/MATH], what is the expected payoff?
My attempt: The payoff is [MATH]2^N[/MATH] if [MATH]2^N\leq X\implies N\leq\log_2X[/MATH]. And it's [MATH]X[/MATH] otherwise. Therefore the expected value is:
[MATH]\sum_{N=0}^{\lfloor\log_2X\rfloor}2^N(0.5)^N\times0.5 + \sum_{N=\lfloor\log_2X\rfloor+1}^{\infty}X(0.5)^{N+1} \\=\frac{\lfloor\log_2X\rfloor+1}{2}+X\big(0.5^{\lfloor\log_2X\rfloor+2}+0.5^{\lfloor\log_2X\rfloor+3}+\ldots\big) \\=\frac{\lfloor\log_2X\rfloor+1}{2}+0.5^{\lfloor\log_2X\rfloor+1}X[/MATH]
Are the approach and solution correct?
My attempt: The payoff is [MATH]2^N[/MATH] if [MATH]2^N\leq X\implies N\leq\log_2X[/MATH]. And it's [MATH]X[/MATH] otherwise. Therefore the expected value is:
[MATH]\sum_{N=0}^{\lfloor\log_2X\rfloor}2^N(0.5)^N\times0.5 + \sum_{N=\lfloor\log_2X\rfloor+1}^{\infty}X(0.5)^{N+1} \\=\frac{\lfloor\log_2X\rfloor+1}{2}+X\big(0.5^{\lfloor\log_2X\rfloor+2}+0.5^{\lfloor\log_2X\rfloor+3}+\ldots\big) \\=\frac{\lfloor\log_2X\rfloor+1}{2}+0.5^{\lfloor\log_2X\rfloor+1}X[/MATH]
Are the approach and solution correct?