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Suppose we have 2 i.i.d random variables $X, Y \sim Ber(0.5)$. What is the image of $\frac{X}{Y}$? Is $\frac{X}{Y}$ a random variable?

Clearly, $I(\frac{X}{Y})$ contains 0 and 1 with $P(\frac{X}{Y} = 0) = P(\frac{X}{Y} = 1) = 0.25$.

What about $+\infty$ when $\frac{X(\omega)}{Y(\omega)} = \frac{0}{0}$ or $\frac{1}{0}$? Is it in the image of $\frac{X}{Y}$? If it is, what is the expected values of $\frac{X}{Y}?$

I asked this question because Student's t distribution is the distribution of the random variable $T = \frac{R_1}{R_2}$ in which $R_1 \sim N(0,1)$, $R_2 \sim \chi^2_n$, and $R_1, R_2$ are independent. $0 \in I(R_2)$ when $n \geq 2$. I want to know if $+\infty$ is something in the image of the random variable $T$ or it is simply "undefined"?

I also want to know what is the difference between the first situation $\frac{X}{Y}$ and the second situation with Student's t distribution.

whuber
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    Although it is possible to use the "extended reals" to define $1/0=\infty$ and $-1/0=-\infty,$ expressions like $0/0$ are never inherently defined. To see how to analyze the continuous case, study https://stats.stackexchange.com/a/299765/919 for instance. – whuber Jun 15 '22 at 22:40
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    Thank you so much! –  Jun 15 '22 at 23:50

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