I know that the ratio of two independent normal random variables is a Cauchy random variable.
The ratio of WHAT type of independent random variables is normal?
2025-01-13 02:57:19.1736737039
The ratio of WHAT type of independent random variables is normal?
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I'll address the case $X$ and $Y$ are i.i.d. (otherwise it's trivial).
Then we can't have $Z=X/Y$ s.t. $E(|Z|)<\infty$ and $p_Z(0)>0$. Suppose otherwise: $$E(|Z|)=E(|X|)E\big(\frac 1 {|Y|}\big)<\infty$$ which implies that $p_Y(0)=0$. On the other hand: $$0<p_{Z}(0)=\int|y|p_X(0)p_Y(y)dy=0$$ Contradiction.