The following is a question from a past paper for one of my university statistical inference modules, and I know how to use the formula for each the max/min, but
Assume that the sample $X_1, X_2, . . . , X_n$ comes from a continuous distribution with cumulative distribution function $F(x)$ and probability density function $f(x)$. Show that the probability density functions of the maximum $(z)$ and minimum $(w)$ of the sample are respectively given by:
$g(z) = nf(z)[F(z)](n−1)$
and
$h(w) = nf(w)[1 − F(w)](n-1)$
NB: the (n-1)'s above are supposed to be indices
Assume that the sample $X_1, X_2, . . . , X_n$ comes from a continuous distribution with cumulative distribution function $F(x)$ and probability density function $f(x)$. Show that the probability density functions of the maximum $(z)$ and minimum $(w)$ of the sample are respectively given by:
$$g(z) = nf(z)[F(z)]^{n−1}$$
and
$$h(w) = nf(w)[1 − F(w)]^{n-1}.$$
If someone could provide a proof for one, or both, that would be much appreciated.
Thank you
