1
$\begingroup$

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

$\endgroup$
1
$\begingroup$

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.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.