Conditional expectation and inequality

Question

This should be relatively easy question but my math skills are rather limited.

Assume a that i is uniformly distributed between $\ [0,3] $

Also, assume that there is a price such that:

$$p \ge E[v * i | w * i \le p]$$

v and w are two unknown parameter with $\ w \gt v $. How do I calculate the conditional expectation of this formula, and how I handle the fact that the price parameter conditions i?

Please use MathJax to format your question. – iiivooo Apr 05 '16 at 23:48 — iiivooo, Apr 05 '16 at 23:48

score 1 · Accepted Answer · answered Apr 06 '16 at 00:05

1

Assuming that $w,v$ are deterministic and for simplicity both $w,v>0$.

If $I \sim U(0,3)$, then $vI \sim U(0,3v)$ and $P(vI|wI \leq p) \sim U(0,\frac{vp}{w})$. Therefore $$\mathrm E[vI|wI \leq p] = \frac{vp}{2w}.$$ From this you can see the contidition under which such $p$ exists.

answered Apr 06 '16 at 00:05

iiivooo

1,208

Thank you for trying to answer the question, but I am still confused about the general approach you took here. Sorry. – user1738753 Apr 06 '16 at 00:17
Can you derive distribution of $vI $ and then use the Bayes' theorem to derive the distribution of $P(vI|wI \leq p)$ ? – iiivooo Apr 06 '16 at 00:27
No, I can't. I am sorry. – user1738753 Apr 06 '16 at 00:33
Actually just got it – user1738753 Apr 06 '16 at 00:46
@user1738753 wonderful, glad to help! – iiivooo Apr 06 '16 at 02:21

Conditional expectation and inequality

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