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Given two distributions $Y$ and $X$ with joint probability density function $f(x,y)$, and letting $U=Y/X$, then I would write the cumulative distribution function of $u$ as:

\begin{equation} P(U \le u) = P(Y \le uX | X > 0) + P(Y \le uX | X < 0) \end{equation}

However, all texts write it as:

\begin{equation} P(U \le u) = P(Y \le uX | X > 0) + P(Y \ge uX | X < 0) \end{equation}

i.e. a difference in the 2nd term on the RHS. I realise this term is in the region of $X < 0$ and hence some rearrangement is required, but I cannot see how precisely how this is done.

Can someone please show this?

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    $\begingroup$ Multiplication by a negative number changes the direction of the inequality sign. $\endgroup$
    – Andrew
    Apr 8, 2023 at 17:34
  • $\begingroup$ Doh! Thank you. $\endgroup$
    – JohnRoper
    Apr 8, 2023 at 17:41

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