# Maximum of log-normal random variable

How can I determine the maximum of n iid (independent and identically distributed) random variables? The random variable follows log-normal distribution.

Let $X_1,X_2,\ldots,X_n$ be iid random variables, and let $$X=\max\{X_1,\ldots,X_n\}.$$ Then $$P(X\leq x) = P(X_1\leq x,X_2\leq x,\ldots,X_n\leq x).$$ Why? Because if the maximum is less than or equal to some $x$, then each of the variables must certainly be less than or equal to $x$ as well. Vice versa, if each of the variables are less than or equal to $x$, then their maximum must be as well.
Since the random variables are iid, we have $$P(X\leq x) = P(X_1\leq x,\ldots,X_n\leq x) = P(X_1\leq x)^n.$$
In conclusion, the probability distribution function of the maximum can be found from the probability distribution function of one of the variables (e.g. the first one) to the $n$th power. You just need to find $P(X_1\leq x)^n$, where $X_1$ is log-normal.