Transforming uniform variables

[wtrow]

If X~uniform(0,1), then what is the pdf of 1/X?

I'd assume it would be the same thing, since X=1 for 0<x<1, so 1/X would be 1/1. Not sure if it works that way, so just making sure.

 

[tkhunny]

What?

Uniform: $\displaystyle P(X < x) = \int_{-\infty}^{x}\;dt = \int_{0}^{x}\;dt = x$

Your task: $\displaystyle P(Y < y = 1/x) = ??$

Your difficulty is this: x = 1 ==> y = 1. That's the easy part. x = 0 ==> y = ?? That's the hard part.

Try again.

 

[royhaas]

Let $\displaystyle Y = 1/X$. Then consider the equivalent events $\displaystyle Y <= y$ and $\displaystyle 1/X <= y$ and $\displaystyle X >= 1/y$.

 

[tkhunny]

Listen to royhaas. I always manage to confuse myself.