i have a Normal(0,1)=X. and (X_{1},....X_{20}). I have to calculate the distribution of $T=\sqrt{|Z|}$ with Z= $\dfrac{1}{20} \sum_{1}^{20}X_{i}$ and his average. I have done this, but Im not very sure:\
Z= $\dfrac{1}{n} \sum_{1}^{20}X_{i} =N(0,1)$\
$P(T=x)=P(\sqrt{|Z|}=x)=P(Z=x^2)+P(z=-x^2)$ =$2f(x^2)=\dfrac{2}{\sqrt{2\pi}}exp({\dfrac{-x^{4}}{2}})$ $x\geq 0$\ it looks a bit strange no?? thanks!
Assuming $X_i$ are independent identically distributed standard normal random variable start by noticing that the mean is a Gaussian random variable with zero mean: $$ Z = \frac{1}{n} \sum_{i=1}^n X_i \sim \mathcal{N}\left(\mu=0,\sigma^2=\frac{1}{n}\right) $$ Then compute the cumulative distribution function of $T= \sqrt{|Z|}$ for $t>0$: $$ F_T(t) = \Pr\left(T\leqslant t\right) = \Pr\left(-t^2 \leqslant Z \leqslant t^2\right) = \Phi\left(\frac{t^2}{\sigma}\right) - \Phi\left(-\frac{t^2}{\sigma}\right) = 2\Phi\left(\sqrt{n} t^2\right)-1 $$ Hence the density $f_T(t) = F_T^\prime(t)$ is $$ f_T(t) = 4 \sqrt{n} t \phi(\sqrt{n} t^2) \left[ t \geqslant 0 \right] = \sqrt{\frac{8 n}{\pi}} \cdot t \cdot \exp\left(-n \frac{t^4}{2}\right) \left[ t \geqslant 0 \right] $$