Consider ($X_1, X_2,\cdots, X_n$) a random sample of a population of $X$ as a Weibull distribution of parameters $(0, \delta, 2), \delta\in\mathbb R^+$, (in short , $X \sim W (0, \delta, 2)$ where $X$ has distribution function $F(x) = 1 - e^{-\left(\frac x\delta\right)^2}$. Consider $$\overline{X}_n = \frac1n\sum\limits_{i = 1}^n X_i\text{ and }N_n = \min(X_1, X_2, \cdots, X_n)$$
I already calculate $E(X)$ and $V(X)$ but now I'm trying to calculate $E(N_n)$ and $V(N_n)$. How can I calculate $E(N)$ and $V(N)$? I have no idea...
The solution is
$$E(N_n) = \frac{\sqrt\pi}2\frac\delta{\sqrt n}\text{ and }V(N_n) = \left(1 - \frac\pi4\right)\left(\frac\delta{\sqrt n}\right)^2 = \left(1 - \frac\pi4\right)\frac{\delta^2}n.$$
For any $t>0$, we have $$ \{\min(X_1,\ldots,X_n)>t\} = \bigcap_{i=1}^n\{X_i>t\}, $$ and so $$ \mathbb P(N_n > t) = \mathbb P\left(\bigcap_{i=1}^n\{X_i>t\}\right). $$ Since the $X_i$ are independent, $$\mathbb P\left(\bigcap_{i=1}^n\{X_i>t\}\right) = \prod_{i=1}^n \mathbb P(X_i>t). $$ Since the $X_i$ all have $W(0,\delta,2)$ distribution, $$ \prod_{i=1}^n \mathbb P(X_i>t) = \mathbb P(X_1>t)^n = (1-F(t))^n = e^{-\left(\frac x\delta\right)^{2n}}. $$ It follows that $$ \mathbb E[N_n] = \int_0^\infty (1-F_{N_n}(t))\ \mathsf dt = \int_0^\infty e^{-\left(\frac x\delta\right)^{2n}}\ \mathsf dt = \delta\cdot \Gamma \left(1+\frac{1}{2 n}\right), $$ and $$ \mathbb E[N_n^2] = \int_0^\infty 2t(1-F_{N_n}(t))\ \mathsf dt = \int_0^\infty 2te^{-\left(\frac x\delta\right)^{2n}}\ \mathsf dt = \delta^2 \Gamma \left(1+\frac{1}{n}\right). $$ Therefore \begin{align} \operatorname{Var}(N_n) &= \mathbb E[N_n^2] - \mathbb E[N_n]^2\\ &= \delta^2 \Gamma \left(1+\frac{1}{n}\right) - \left(\delta\cdot \Gamma \left(1+\frac{1}{2 n}\right)\right)^2\\ &= \delta ^2 \left(\Gamma \left(1+\frac{1}{n}\right)-\Gamma \left(1+\frac{1}{2 n}\right)^2\right) \end{align}