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$$\begin{array}{|c|c|c|c|c|} \hline X&0&200&400&800&1000&2000\\ \hline \mathbb{P}&0.4&0.2&0.1&0.1&0.1&0.1\\ \hline \end{array}$$ The principle of zero utility formula:

\begin{equation} \mathbb{E}u(x+\pi(X)-X)=u(w). \end{equation}

I am given $u(x)=\sqrt{x}$, $w=10000$. Now

\begin{equation} \mathbb{E}\sqrt{10000+\pi(X)-X}=\sqrt{10000}. \end{equation}

I know I cannot just simply square everything and get my result. What is the way to get $\pi(X)$? I found similar question (Quadratic Utility Function) and understand the whole procedure with quadratic utility function.

Karagum
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    As you mentioned, direct calculation is not the right way to approach this problem. I don't know the answer, but you could try finding an MGF for X. In addition, $\sqrt{1 + X}$ has a nice Taylor series. So you can use that Taylor series and the MGF to get a quick estimate. – Pavan C. Apr 14 '21 at 19:19
  • @PavanC. You are right, I haven't thought of Taylor series. Thank you! – Karagum Apr 14 '21 at 19:20

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