A random variable constructed by sampling and maxing

Question

Suppose I have a continuous random variable $X$.

Now I define a new random variable $Y$ by sampling $X$ $n$ times and then taking the max of the sample. Assume the samples are iit.

$$ Y = max(X_{1}, ... , X_{n}) $$

1. Is it possible to say something about $Y$ in terms of $X$, generally?

2. Suppose that $X$ is normal. Now what can I expect from $Y$?

Answer 1

You can express the CDF of $Y$ quite easily,

$$ F_Y(y) = \mathbb{P}(Y \leq y) = \prod\limits_i^n\mathbb{P}(X_i \leq y) = (\mathbb{P}(X_i \leq y))^n$$

This is true because $max(X_1,..,X_n) \leq y \Longleftrightarrow X_i \leq y$ for all $i$.