Question

Suppose u1, u2, ..., un are independent random variables and for every i = 1, ..., n, ui has a uniform distribution over [0, 1]. define z = min(u1, ..., un). find the
c.d.f. and the p.d.f of z.

Answer 1

$Z=U_{(1)}=\min\{U_1,\ldots,U_n\}$

has CDF

$F_Z(z)=1-(1-F_{U_i}(z))^n$

where $F_{U_i}(u_i)$ is the CDF of $U_i$. Since $U_i$ are iid. with the standard uniform distribution, we have

$F_{U_i}(u_i)=\begin{cases}0&\text{for }u_i$

and so

$F_Z(z)=1-(1-F_{U_i}(z))^n=\begin{cases}0&\text{for }z$

Differentiate the CDF with respect to $z$ to obtain the PDF:

$f_Z(z)=\dfrac{\mathrm dF_Z(z)}{\mathrm dz}=\begin{cases}n(1-z)^{n-1}&\text{for }0$

i.e. $Z$ has a Beta distribution $\beta(1,n)$.