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Prof.~David Draper \\
Department of Applied Mathematics and Statistics \\
University of California, Santa Cruz
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\textbf{\large AMS 131: Quiz 8}
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Name: \underline{\hspace*{5.85in}}
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Someone offers you the possibility to play a gambling game with the following rules. First, you decide how much money you're willing to put at risk in this game: this amount --- let's call it $A$ --- is referred to as your \textit{stake} (all the monetary quantities are in dollars in this problem). Having chosen your stake, you're allowed to bet any amount $0 \le B \le A$ (thus, as a decision problem, for any fixed value of $A$, your possible actions in this situation correspond to values of $B$). If you win the bet, which occurs with probability $0 < p < 1$, your stake becomes $( A + B )$; if you lose, it becomes $( A - B )$, and this (of course) occurs with probability $( 1 - p )$; and (crucially) $p$ is known to you. Let $X$ denote the value of your stake after the gamble has occurred, and suppose that you agree with Daniel Bernoulli that a reasonable utility function is $U ( x ) = 1 + \log ( x )$.
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\item[(a)]
Write out the probability mass function (PMF) for $X$.
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\item[(b)]
Work out your expected utility $E [ U ( X ) ]$ in this game, as a function of $A, B$ and $p$.
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\item[(c)]
Intuitively, what should you do (i.e., what value of $B$ should you choose) if $p < \frac{ 1 }{ 2 }$? Explain briefly.
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\item[(d)]
Show that when $p \ge \frac{ 1 }{ 2 }$ your expected utility is maximized with the choice $B = ( 2 \, p - 1 ) \, A$. Is this answer intuitively reasonable? Explain briefly.
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