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Let $X_1\sim N(3,\sqrt{5})$ and $X_2\sim\chi ^2(15)$. Assume $X_1,X_2$ to be independent.

Determine the distribution of $$Z := \sqrt{\frac{3}{X_2}}(X_1-3) $$

I don't understand what to do here and such a problem raises a question: Suppose $X_1,\ldots ,X_n$ are independent and of normal or chi square distribution and we take an Arbitrary combination of them, can we still determine its distribution?

In other words, I'm not exactly interested in what the distribution of $Z$ is, rather, how we arrive at determining it.

Observations: $$\frac{X_1-3}{\sqrt{5}}\sim N(0,1) $$ also $\sqrt{\frac{1}{X_2/15}} = \sqrt{\frac{1}{\chi ^2(15)/15}}$.

$$\sqrt{\frac{1}{\chi ^2(15)/15}}\overset{?}\sim \sqrt{F(1,15)} $$ so $Z\sim \sqrt{F(1,15)}\cdot N(0,1)$?? :D looks dubious

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As you have observed, $$ Y = \frac{X_1 - 3}{\sqrt{5}} \sim \mathcal{N}(0, 1) $$ which implies $$ Y^2 \sim \chi^2(1) $$ Therefore, $$ Z^2 = \frac{Y^2 / 1}{X_2 / 15} \sim F(1, 15) \tag{$1$} $$ For $(1)$, please refer to the wiki on F-ditribution.

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