Maximize Your Gain 2：赢 c 或输 c

Maximize Your Gain 2

专题: Probability / 概率
难度: L4
来源: QuantQuestion

题目详情

Under the same assumptions as in Problem 1.9, the rule of the game is changed, as follows: if $\mathrm{U} > \mathrm{c}$ an amount of c is won, but otherwise the amount c is lost.

a. Characterize the value of c that maximizes the expected gain, G(c). b. Can the game be unfavorable, i.e., can (even) the maximum expected gain become negative? c. How large is the maximum expected gain for an exponential rv with rate A? Compare this to the solution of the previous problem; explain the difference.

解析

收益为 $G(c)=c[1-2F(c)]$ 。一阶条件\n\n $\nG'(c)=1-2F(c)-2c f(c)=0\\Rightarrow \\boxed{1-2F(c)=2c f(c)}.\n$ \n\n指数分布 $F(c)=1-e^{-\\lambda c}$ 下方程为\n\n $\n2e^{-\\lambda c}(1-\\lambda c)=1,\n$ \n\n数值解 $c\\approx 0.315/\\lambda$ ，最大期望约 $0.145/\\lambda$ 。