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**There's this problem that I've been trying to solve. I know the solution for it now but my initial attempt at a solution was wrong and I can't seem to figure out the mistake with my reasoning. I'd appreciate some help with figuring this one out.**

1. Homework Statement

I have a set of random variables drawn independently from a distribution. And a new random variable.

1. Homework Statement

I have a set of random variables drawn independently from a distribution. And a new random variable.

[tex]Z = min\{X_1, X_2, ... X_N\}[/tex].

Each [itex]X_i[/itex] has the pdf [itex]f_X(x)[/itex] and CDF [itex]F_X(x)[/itex]

What I want to do is to find the CDF (and then the PDF) of Z.

**3. The Attempt at a Solution**

So here's what I tried first.

[tex]P(Z<z) = P((\exists i\ s.t\ X_i < z) \cap (X_j > z\ \forall j \neq i)) [/tex]

[tex] P(Z<z) = \left(\sum_{i=1}^{N}P(X_i < z)\right) \left( \sum_{j=1, j\neq i}^{N}P(X_j < z) \right)[/tex]

[tex] P(Z<z) = N(N-1)F_X(z)(1-F_X(z))[/tex]

But I know this is wrong because I did some research and I know that the correct (and easier) way to do it is to find [itex]P(Z > z)[/itex]. The actual answer is [itex]1 - (1 - F_X(z))^N [/itex].

Can someone help me find the flaw in my reasoning?