Can I get help on the following HW question:
If X is a standard Cauchy Random Variable then 1/X is also a standard Cauchy Random Variable. How can it be proved?
Is it correct to say that: P(1/X<x) = P(X>1/x) = 1 - P(X<1/x) for x>0?
What about if x<0
Can I get help on the following HW question:
If X is a standard Cauchy Random Variable then 1/X is also a standard Cauchy Random Variable. How can it be proved?
Is it correct to say that: P(1/X<x) = P(X>1/x) = 1 - P(X<1/x) for x>0?
What about if x<0
You seem to be on the right track. I would start with defining $\displaystyle Y = \frac{1}{X}$, and begin with $\displaystyle F_{Y}(y) = P \left (Y \le y \right ) = P\left (\frac{1}{X} \le y \right ) = P \left (X \ge \frac{1}{y} \right ) = 1 - F_{X}(1 / y)$, then take the derivative and see if you recognize it.
I'm not sure this works, since g(X) = 1/X is not monotone over the whole real line, and moreover is undefined for part of the support of X. The method DOES give the right answer, but I think if you are going to use it, you should handle to problem point X = 0. I think it's easier just to use work directly with the cdf.