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Thread: Distribution function of the minimum of a random sample

  1. #1
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    Distribution function of the minimum of a random sample

    I'm trying to understand the distribution function of the minimum of a random sample.

    Distribution function of the minimum of a random sample-img_1999.jpeg

    x_1,x_2,...,x_n are i.i.d

    m_n=min(x_1,x_2,...,x_n)
    M_n=max(x_1,x_2,...,x_n)

    We know that P(M_n<=z)= F_max (z)=(F(z))^n

    My attempt to understand this:

    P(m_n<=z)=P(-M_n<=z)
    =P(M_n>=-z)
    =1-P(M_n<-z)

    After this I'm really stuck. Appreciate your help.
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  2. #2
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    Re: Distribution function of the minimum of a random sample

    You've got an iid set of samples of some random variable $X_n$ with a minimum $m=\min(X_n)$

    $P[\text{m is minimum}] = P[X_n \geq m,~\forall n] = \prod \limits_n P[X_n \geq m] = \prod \limits_n (1-F_X(m)) = (1-F_X(m))^n$

    similarly

    $P[\text{M is maximum}] = P[X_n \leq M,~\forall n] = \prod \limits_n P[X_n \leq M] = \prod \limits_n (F_X(M)) = (F_X(M))^n$
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