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Math Help - Stats, c.d.f, maximums, random variables

  1. #1
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    Question 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

    but am not confident about this, is any of this ok . . ???

    thanks for any help
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  2. #2
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    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 . . . . . . . .
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  3. #3
    MHF Contributor matheagle's Avatar
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    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
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