Product & Ratio's of 2 Random Variables

Question

I'm interested to know whether it's the case that for random variables $X$ and $Y$ whether or not the ratio of $X$ and $Y$ can be computed as the product of $X$ and $1/Y$. That is,

Is

$\frac{X}{Y} \stackrel{?}{=} X \cdot \frac{1}{Y}$

a true (or false) statement.

Thanks for your help in these matters.

Answer 1

Let $X$ and $Y$ be two random variables. $\frac XY$ is defined as a random variable $\omega\mapsto\frac{X(\omega)}{Y(\omega)}$ provided that $Y(\omega)\ne0$ for each $\omega\in\Omega$ (recall that the quotient of two measurable functions is a measurable function). The product of two random variables is defined accordingly.

$\frac1Y$ is given by $\omega\mapsto\frac 1{Y(\omega)}$ and $X\cdot\frac1Y$ is given by $\omega\mapsto X(\omega)\cdot\frac1{Y(\omega)}$. Hence, $\frac XY=X\cdot\frac1 Y$.

Answer 2

By definition, "$\frac{X}{Y}$" is the random variable obtained as follows: determine the random values for $X$ and $Y$, then divide them. How could that not be equivalent to determining the random values for $X$ and $Y$, then computing $X\cdot\frac{1}{Y}$? After all, these arithmetic operations are equivalent.

(All of this assumes, of course, that $Y\neq0$.)