How to show that something has the same probability distribution function as something else?

Question

I understand the logic behind what I'm about to ask, but I'm not sure exactly how to write it mathematically. I have a random variable $X$ with probability density function:

$$ f_X(x)= \begin{cases} x^{-2} & x\ge 1\\ 0 & x<1 \end{cases} $$

and therefore a probability distribution function:

$$ F_X(x)= \begin{cases} 1-\frac{1}{x} & x\ge 1\\ 0 & x<1 \end{cases} $$

Then I need to show that if $$ is uniformly distribution between $0$ and $1$ (i.e. $∼(0,1)$) and $=(1−)^{−1}$, then $$ has the same distribution function as $X$.

I get that $0<\frac{1}{x}<1$, but I'm not sure how to use this to show the above - let alone how to show it in a mathematically sound way.

Answer 1

Hints:

• If $y=\frac{1}{1-w}$, then can you find $w$ in terms of $y$? Let's call this $w=g(y)$
• Is $y$ increasing or decreasing when $w$ is positive and increasing?
• What is $\mathbb P(W \le w)$ in terms of $w$ when $w \le 0$ or $0 \lt w \lt 1$ or $1 \le w$?
• What is $\mathbb P(W \le g(y))$ in terms of $g(y)$? Over what ranges of $y$?
• What is $\mathbb P(Y \le y)$ in terms of $y$? Over what ranges of $y$?

• Does this match $F_X(x)$?