Consider a random samples $X_1,X_2,..,X_n$ from a variables with density function $$f_X(x)=2\lambda\pi xe^{-\lambda\pi x^2} \ \ \ \ \ \ \ x>0$$
I have shows that for $i=1,..,n$, $X^2_i\sim \text{Gamma}(1,\frac{1}{\pi\lambda})$
Now show that the density function of $T=\frac{\pi}{n}\sum_{i=1}^{n} X^2_i$ is $$f_T(t)=\frac{n^n\lambda^n t^{n-1}e^{-n\lambda t}}{\Gamma(n)} \ \ \ \ \ \ \ \ t>0$$
I tried to solve this in a similar way to showing $X^2_i\sim \text{Gamma}(1,\frac{1}{\pi\lambda})$. I let $$Z=\sum_{i=1}^{n} X^2_i \ \ \ \ \ \text{such that} \ \ Z\sim \text{Gamma}(1,\frac{1}{\pi\lambda})\Rightarrow f_Z(z)=\lambda\pi e^{-\lambda\pi z}$$ Therefore $$T=\frac{\pi}{n} Z$$ which is monotonic over $z>0$. Hence \begin{align*} f_T(t)&=f_Z(t)\Big|\frac{dx}{dt}\Big| \\ &=\lambda\pi e^{-\lambda\pi t}\Big|\frac{n}{\pi}\Big| \\ &=\lambda n e^{-\lambda\pi t} \ \ \ \ t>0 \\ \end{align*} But this is clearly not the density required. Where have I made a mistake?
Some Hints:
Let $X_i$ be a Gamma distribution r.v. with parameters $(a_i,b)$: $$X_i\sim \text{Gamma}(a_i,b)$$
One can show (by moment generating functions or direct derivation of pdf) that: $$\sum_i^nX_i \sim Gamma(\sum_i^n a_i,b)$$
One may also show that by the scaling property of a Gamma distribution: $$c \ X_i \sim Gamma(a,c \cdot b)$$
Can you take it from here?