Chapter 7, Problem 80SE

a.

To determine

Show that cumulative distribution functions of these two order statistics are $F_{X_{\left(1\right)}}\left(t\right)=1-{\left[1-F\left(t\right)\right]}^n$, $F_{X_{\left(n\right)}}\left(t\right)={\left[F\left(t\right)\right]}^n$.

b.

To determine

Prove that the probability distributions of $X_{\left(1\right)}$ and $X_{\left(n\right)}$ are $f_{X_{\left(1\right)}}\left(t\right)=n{\left[1-F\left(t\right)\right]}^{n-1}f\left(t\right)$ and $f_{X_{\left(n\right)}}\left(t\right)=n{\left[F\left(t\right)\right]}^{n-1}f\left(t\right)$ when X is continuously with probability density function $f\left(x\right)$.

c.

To determine

Show that $P\left(X_{\left(n\right)}=1\right)=1-{\left(1-p\right)}^n$ and $P\left(X_{\left(1\right)}=0\right)=1-p^n$.

d.

To determine

Derive the probability density functions of $X_{\left(1\right)}$ and $X_{\left(n\right)}$.

e.

To determine

Derive the cumulative density functions of $X_{\left(1\right)}$ and $X_{\left(n\right)}$.

Derive the probability density functions of $X_{\left(1\right)}$ and $X_{\left(n\right)}$.