Hard Probability Question

**WeeG** · August 3rd 2010, 12:24 AM

Hi Guys, I have quite a hard question, I am attaching 2 picture, one of the question and 1 of my attempt to solve the first part, I must have done something wrong, either from probability point of view or calculus...

any ideas for "a" and "b" (which I have no clue about) will be appreciated.

Thank you !!

**SpringFan25** · August 3rd 2010, 07:55 AM

a)

i have not checked your solution in detail, but i would point out that

$e^{20-lnr} = e^{20}e^{-lnr} = e^{20}r^{-1}$

i think that makes your integral a bit more do-able

b) The profit (X) appears to be

$X = 1000 - \frac{1000}{T}$

(Note that the minimum value of T is 20, so the minimum value of X is 950)

Start by finding the CDF of X
$P(X<x) = P(1000-\frac{1000}{T} < x )$

$P(X<x) = P(\frac{1000}{T} > 1000 - x )$

$P(X<x) = P(\frac{T}{1000} < \frac{1}{1000 - x} )$

$P(X<x) = P(T < \frac{1000}{1000 - x} )$

$P(X<x) = F_T(\frac{1000}{1000-x})$

$P(X<x) = 1-exp \left( 20 - \frac{1000}{1000-x} \right) , x \geq 950$

Assuming my algebra was correct (check!), differentiate to obtain the desity of x.

**WeeG** · August 3rd 2010, 08:12 AM

thank you!! I did what you suggested, I think I got the correct result !

**SpringFan25** · August 3rd 2010, 08:15 AM

woo

i also edited my original post while you were typong that, has more info on part b now.

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