# Thread: Distribution function of the minimum of a random sample

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

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

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.

2. ## 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

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