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<0\\u_i&\text{for }0\le u_i<1\\1&\text{for }u_1\ge1\end{cases}`

and so


`F_Z(z)=1-(1-F_(U_i)(z))^n=\begin{cases}0&\text{for }z<0\\1-(1-z)^n&\text{for }0\le z<1\\1&\text{for }z\ge1\end{cases}`

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


`f_Z(z)=(\mathrm dF_Z(z))/(\mathrm dz)=\begin{cases}n(1-z)^(n-1)&\text{for }0<z<1\\0&\text{otherwise}\end{cases}`

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