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In the notes I'm working through, distributions are often "divided" by other distribution, and while I sort of understand what is meant, i would rather a rigorous explanation.

Let me provide an example:

$U$~$N(3,16)$

$V$ ~$\chi_{9}^{2}$

U and V are independent random variables. Find $P(U-3<4.33\sqrt{V})$

$P(U-3<4.33\sqrt{V})$= $P(N(3,16)-3<4.33\sqrt{V})$= $P(N(0,16)<4.33\sqrt{V})$ =$P(4 \times N(0,1)<4.33\sqrt{V})$

=$P(\frac { N(0,1)}{\sqrt{V}} <\frac{4.33}{4})$

=$P(\frac {N(0,1)}{\sqrt{\chi_{9}}} <\frac{4.33}{4})$

=$P(3\times\frac {N(0,1)}{\sqrt{\chi_{9}}} <3\times\frac{4.33}{4})$

=$P(\frac {N(0,1)}{\sqrt{\frac{\chi_{9}}{9}}} <3\times\frac{4.33}{4})$ =$P(t_{9} <3\times\frac{4.33}{4})$

(where $t_{9}$ denotes the t distribution with 9 degrees of freedom)

I understand that "If $X_1,...,X_n$ are a random sample from $N(\mu,\sigma^2)$ then $\frac{\overline{X}-\mu}{\frac{s}{\sqrt{n}}}$ is distributed $t_{(n-1)}$ where $s=\sqrt{\frac{1}{n-1}\sum(X_i-\overline{X})^2}$.

But what does it mean to divide $N(0,1)$ by $\chi_{9}$?? why is $\frac {N(0,1)}{\sqrt{\frac{\chi_{9}}{9}}}$ equivalent to $t_{9}$??

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    $\begingroup$ The notation is truly awful ... (and not your fault, since it seems you're quoting somebody). $\endgroup$ – whuber May 17 '19 at 14:26
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    $\begingroup$ @stochasticmrfox One further problem is that their notation seems to conflate the chi-square with the chi, a recipe for confusion. $\endgroup$ – Glen_b -Reinstate Monica May 17 '19 at 16:58
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They're not dividing distributions. They're using a lot of shortcut notation. The piece that you're confused with appears to be this one: $$\frac { N(0,1)}{\sqrt{\frac {\chi_9} 9}} $$

What they mean here is the new variable: $$X=\frac{\xi}{u}$$ where $\xi$ is from standard normal distribution and $u$ is from $\chi$ distribution with 9 degrees of freedom scaled by the degrees of freedom. So, this variable $X$ will have its own distribution, which happens to be Student t distribution.

How do they know that it's Student t? They must know the history: it's how Gosset derived this distribution. In fact, I think this is a homework, and it was specifically designed to test whether you can spot the possibility to reduce the problem to the Student t distribution by noticing that it can be re arranged to have a ratio of normal and square root of chi-squared random variables.

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