# 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 $$f_{Z}(z)$$.

Thanks so much in advance for all your help.

## 1 Answer

The input needed here is the hard-to-immediately-grasp idea that the minimum order statistic exceeding a threshold is equivalent to having all the random variables of the sequence being greater than that threshold for non-negative random variables (such as yours). That is

$$P(Z>z) = P(\min(X_1,X_2,\dots, X_n)>z) = P(X_1>z,X_2 >z, \dots, X_n>z)$$

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
• Yes, indeed, that is correct! – Easymode44 Nov 11 '18 at 19:37
• a huge hyper mega thank you! As soon as I can Ill vote you up <3 – asd123 Nov 11 '18 at 19:39
• @Easymode44 Minor comment: $P(Z>z)=1-P(Z \leq z)$. – Aditya Dua Nov 11 '18 at 22:22
• @AdityaDua thanks! Edited! – Easymode44 Nov 11 '18 at 23:23