1. ## Hard Probability Question

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 !!

2. a)

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

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

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

b) The profit (X) appears to be

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

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

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

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

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

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

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

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

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

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

4. woo

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