Thread: Stats, c.d.f, maximums, random variables

1. Stats, c.d.f, maximums, random variables

Hi
k heres the question

Let u1 , u2 , u3 . . . un, . . . be random variables each with uniform distributions on [0,1]. let Mn = max(u1, u2, . . . , un).

Express the event {Mn<=c} in terms of u1, u2 , ... , un and hence determine the cummulative distribution function Mn. ?

can i do something like this {Mn<=c}={max(u1, u2, . . ., un)<=c}=
{max({u1<=c}, {u2<=c}, . . ., {un<=c}}

I found c.d.f of Mn to be =0 if c<=0
=c if 0<c<1
=1 if 1<=c

thanks for any help

2. FMn(c) = p(Mn<=c)=p(max(U1,U2, . . ,Un))=p((U1<=c) and (U2<=c} and . . . .(Un<=c)) = p(U1<=c)p(U2<=c).....p(Un<=c)

fUi(x) = 1 if 0<x<1
0 otherwise

for c<=0 p(Ui<=c)= 0
for 0<c<1 p(Ui<=c)= c
for c>=1 p(Ui<=c)= 1

so

FMn(c)= 0 if c<=0
c^n if 0<c<1
1 if c>=1

is this right . . . . . . . .

3. Looks good

$P(M_n\le c)=P(U_1\le c,....,U_n\le c)=P(U_1\le c)\cdots P(U_n\le c)$

If c<0, you have zero
If c>1 you will have one.
and if $0\le c\le 1$ you will have $c^n$