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In the above one can see the detail of this question, I am beginner in this kind of mathematics. I will be very greatful if any one can help me to solve them.

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Have you tried to look at the variances of each of those distributions? – Chinny84 Feb 19 '14 at 10:31
What part are you having problem with? Do you not know much about probability distributions, about utility functions or about the hypotheses on which decisions are supposed to be based? – dafinguzman Feb 19 '14 at 10:56
No, I have not look at the variances, I am very new in this course if one can help me with just part 1, than I can continue with the other parts. – Kiran Feb 19 '14 at 13:05
Hi Kiran, how did you find my answer? Please comment if you have any questions. Did you know that you can accept answers if you like them by clicking the check mark ✓ next to it. This scores points for you and for the person who answered your question. You can find out more about accepting answers here: How do I accept an answer?, Why should we accept answers?. You currently have 13 questions and 0 accepts. If you accept answers to enough of them you'll be able to upvote as well :) – TooTone Feb 25 '14 at 23:12
up vote 1 down vote accepted

I'm assuming this is an assignment question so I'm just going to try to point you in the right direction (and only on the first part: you said in a comment that this was what you wanted help with).

Do any decision makers with (increasing) utility function agree about preferring risk $X_1$ to $X_2$?

The expectations of $X_1$ and $X_2$ are equal: $E(X_1)=E(X_2)$. So there would be no reason to prefer $X_1$ or $X_2$ on this basis.

The question is asking if there is any increasing utility function $u(x)$ such that $$E(u(X_1)) > E(u(X_2)) \\ \iff \\ \sum_{x=0}^{10}u(x)Pr(X_1=x) > \sum_{x=0}^{15}u(x)Pr(X_2=x) \\ \iff \\ \sum_{x=0}^{10}u(x)\binom{10}{x}0.5^x0.5^{1-x} > \sum_{x=0}^{15}u(x)\binom{15}{x}(1/3)^x(2/3)^{1-x}$$

Note that $X_1$'s probability of success, 1/2, is greater than $X_2$'s probability of success, 1/3. But $X_2$ still has the same expectation as $X_1$ because with $X_2$ the maximum number of successes you can get, 15, is greater than the maximum number of successes you can get for $X_1$, 10. This suggests that if such a utility function exists, it is one which disproportionately rewards a lower number of successes.

The graph below shows the line $y=x$ in black for the utility function $u(x)=x$, corresponding to $E(u(X))=E(X)$. There are two other lines in green and red: if one of these corresponded to a utility function where decision makers would prefer $X_1$ to $X_2$, which one would it be?

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thanks for explaining me everything regarding how to accept the answer as I was not familiar with this before as I am very new user here. Now regarding the question, I was thinking that the decision maker will go for green line, if I am wrong please rectify me in it. Thanks – Kiran Feb 27 '14 at 6:18
No problem explaining, it's usual to advise new users (you also now have enough rep to upvote -- my policy when asking qs is to upvote answers I get which address my question and are useful). Re the decision itself, what I suggest you do is to go into excel (or libreoffice, or use R if you know it), and simply compare and contrast. You'll learn a lot. The green line is $x^2$ (the red line if I remember rightly is $\sqrt{x}$). In Excel you can use BINOMDIST to get binomial probabilities, so put these in one column. Put the $x^2$s in another column. Use SUMPRODUCT to get your expectation. – TooTone Feb 27 '14 at 10:16

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