# Minimum of random variables iid

I have lots of problems trying to solve problems involving maximum and minimum of random variables. For example:

Let $$X_{1},X_{2},...,X_{n}$$ random variables iid and $$f_{X}(x)= \frac{1}{x^{2}}$$ when $$x\geq 1$$ and $$f_{X}(x)=0$$ in other case.

My problem is that I have no idea how to deal with the $$Z=min\{X_{1},...,X_{n}\}$$, I should be able to find $$F_{Z}(z)$$ and $$f_{Z}(z)$$.

$$P(Z>z) = P(\min(X_1,X_2,\dots, X_n)>z) = P(X_1>z,X_2 >z, \dots, X_n>z)$$
From here, you could go by finding the cumulative distribution function by using the fact that $$P(Z>z)=1-P(Z\leq z)$$ and the independence assumption of the sequence. The pdf follows through differentiation.
• Thanks so much for your answer. Lets see if I understood, $P(Z>z)=P^{n}(X_{1}>z)=(\int_{z}^{\infty} \frac{1}{x^{2}}dx)^{n}=\frac{1}{z^{n}}$. Is that correct? – asd123 Nov 11 '18 at 19:20
• @Easymode44 Minor comment: $P(Z>z)=1-P(Z \leq z)$. – Aditya Dua Nov 11 '18 at 22:22