Example. Sequence of uniformly distributed random variables

Let $ \mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n} $ be $ n $ i.i.d. jointly distributed random variables, each uniformly distributed on the interval $ \left[0,1\right] $ . Define the new random variables $ \mathbf{W}=\max\left\{ \mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n}\right\} $

and

$ \mathbf{Z}=\min\left\{ \mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n}\right\} $ .

(a) Find the pdf of $ \mathbf{W} $

$ F_{\mathbf{W}}(w)=P\left(\left\{ \mathbf{W}\leq w\right\} \right)=P\left(\left\{ \max\left\{ \mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n}\right\} \leq w\right\} \right)=P\left(\left\{ \mathbf{X}_{1}\leq w\right\} \cap\left\{ \mathbf{X}_{2}\leq w\right\} \cap\cdots\cap\left\{ \mathbf{X}_{n}\leq w\right\} \right) $$ =P\left(\left\{ \mathbf{X}_{1}\leq w\right\} \right)P\left(\left\{ \mathbf{X}_{2}\leq w\right\} \right)\cdots P\left(\left\{ \mathbf{X}_{n}\leq w\right\} \right)=\left(F_{\mathbf{X}}\left(w\right)\right)^{n} $ .

where $ f_{\mathbf{X}}(x)=\mathbf{1}_{\left[0,1\right]}(x) $ and $ F_{X}\left(x\right)=\left\{ \begin{array}{ll} 0 ,x<0\\ x ,0\leq x<1\\ 1 ,x\geq1 \end{array}\right. $ .

$ f_{\mathbf{W}}\left(w\right)=\frac{dF_{\mathbf{W}}\left(w\right)}{dw}=n\left[F_{\mathbf{X}}\left(w\right)\right]^{n-1}\cdot f_{\mathbf{X}}\left(w\right)=n\cdot w^{n-1}\cdot\mathbf{1}_{\left[0,1\right]}(w). $

(b) Find the pdf of $ \mathbf{Z} $ .

$ F_{\mathbf{Z}}(z)=P\left(\left\{ \mathbf{Z}\leq z\right\} \right)=1-P\left(\left\{ \mathbf{Z}>z\right\} \right)=1-P\left(\left\{ \min\left\{ \mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n}\right\} >z\right\} \right) $$ =1-P\left(\left\{ \mathbf{X}_{1}>z\right\} \cap\left\{ \mathbf{X}_{2}>z\right\} \cap\cdots\cap\left\{ \mathbf{X}_{n}>z\right\} \right)=1-\left(1-F_{\mathbf{X}}(z)\right)^{n}. $

$ f_{\mathbf{Z}}(z)=\frac{dF_{\mathbf{Z}}(z)}{dz}=n\left(1-F_{\mathbf{X}}(z)\right)^{n-1}f_{\mathbf{X}}(z)=n\left(1-z\right)^{n-1}\mathbf{1}_{\left[0,1\right]}\left(z\right). $

(c) Find the mean of $ \mathbf{W} $ .

$ E\left[\mathbf{W}\right]=\int_{-\infty}^{\infty}wf_{\mathbf{w}}(w)dw=\int_{0}^{1}nw^{n}dw=\frac{n}{n+1}w^{n+1}|_{0}^{1}=\frac{n}{n+1}. $