contestada

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.

Respuesta :

LammettHash
[tex]Z=U_{(1)}=\min\{U_1,\ldots,U_n\}[/tex]

has CDF

[tex]F_Z(z)=1-(1-F_{U_i}(z))^n[/tex]

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

[tex]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}[/tex]

and so

[tex]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}[/tex]

Differentiate the CDF with respect to [tex]z[/tex] to obtain the PDF:

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

i.e. [tex]Z[/tex] has a Beta distribution [tex]\beta(1,n)[/tex].

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