I am trying the find the solution to problem 7 b) from Bertsekas, "Introduction to probability". My problem is that I cannot get the CDF of $s = (\frac{1}{t} , \infty)$ (the 1st interval below) correctly
here is my attempt...
The prominent values that the random variables X and $S = \frac{1}{X}$ can split into two intervals as:
$x = (0 , t) $ $\rightarrow$ $S = (1/t , \infty)$
$x = (t , r)$ $\rightarrow$ $S = (0 , 1/t)$
It is illustrated below.
For the 2nd interval $x = (t , r)$ $\rightarrow$ $S = (0 , 1/t)$
$P(S < s) = P(1/X < 1/t) = P(X > t) = 1 - P(X < t) = 1 - \frac{\pi\times t^2}{\pi\times r^2} = 1 - \frac{t^2}{r^2}$
Now for the 1st interval $x = (0 , t) $ $\rightarrow$ $S = (1/t , \infty)$
$P(S > s) = P(S > \frac{1}{t}) = P(\frac{1}{X} > \frac{1}{t}) = P(X < t) = \frac{\pi \times x^2}{\pi \times r^2} $
using the definition $S = \frac{1}{X}$
$P(S > s) = \frac{1}{(sr)^2}$
Does this mean for 1st interval $ \frac{1}{t}< s < \infty$
$F(s) = P(S < s) = 1 - \frac{t^2}{r^2} + \frac{1}{(sr)^2}$
This is illustrated below
I know that this is not correct because F(s) should be monotonically increasing and reach 1 as S tends to infinity.
I dont understand the solution from the book given below.
Please help to explain to me.
Thanks.
There is a bit of sloppy writing, but I'm ignoring this as at the end you come the right conclusions.
For $s > \frac1t$, you correctly come to the conclusion that $P(S > s)=\frac1{s^2r^2}$ Since for $s > \frac1t$ $S$ is continuous, that means $P(S = s) = 0$, so consequently $P(S < s)=1-\frac1{s^2r^2}$, because all those probabilities needs to add up to $1$, which gives the solutuion presented in the book.
Your error is in dragging the term $-\frac{t^2}{r^2}$ into this, which has no basis.