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Consider a random sample $X_1,X_2,..,X_n$ from a variable with density function $$f_X(x)=2\lambda\pi xe^{-\lambda\pi x^2}, \quad x\geq 0$$ I am trying to show that for $i=1,..,n,\ X^2_i\sim \text{Gamma}(1,\frac{1}{\pi\lambda})$

Now I don't really know where to begin with this question. I'm not even sure what $\sum_{i=1}^{\infty} X^2_i$ equals. Apologies for not including an attempt, this question has stumped me.

Attempt:

We want to find $f_Y(y)$ where $Y=X^2$.

First we note that $y=x^2$ is monotonic over $x>0$

Hence \begin{align*} f_Y(y)&=f_X(y)\Big|\frac{dx}{dy}\Big| \\ &=2\lambda\pi\sqrt{y}e^{-\lambda\pi y} \Big|\frac{1}{2\sqrt{y}}\Big| \\ &=\lambda\pi e^{-\lambda\pi y} \ \ \ \ \ y>0 \end{align*} Since $Y_i\sim \text{Gamma}(1,\frac{1}{\pi\lambda})$, we can see that $X^2_i\sim \text{Gamma}(1,\frac{1}{\pi\lambda})$

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  • $\begingroup$ Do you know about Jacobians? Transforms of random variables? Inverse transform method? $\endgroup$ – Tony Hellmuth Jul 1 '18 at 3:25
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    $\begingroup$ You may find it better to read up on the methods available on transformations of random variables. It will give you a better understanding of how to apply them in general! There are plenty of ways to tackle this problem, this will give you a good understanding with examples. On the other hand, here is a similar question with a worked example. $\endgroup$ – Tony Hellmuth Jul 1 '18 at 3:32
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    $\begingroup$ Why does this have so many downvotes? $\endgroup$ – Prime Jul 1 '18 at 4:08
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    $\begingroup$ @Prime: please feel free to upvote hahah perhaps it was because I did not initially have a solution $\endgroup$ – user557493 Jul 1 '18 at 4:10
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    $\begingroup$ +1 for the attempt lol $\endgroup$ – Prime Jul 1 '18 at 4:11

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