CDF and PDF of Maximum Draw

Question

I'm reading some lecture notes on order statistics and I was hoping to get some clarification.

The highest draw of $z$ draws from the continuous distribution F is given by

1)$$ F^z(x)$$

Why is it $F(x)$ to the power of z? What does it look like when you multiply a distribution by itself? What would this look like in the case of a uniform distribution for instance?

and,

The pdf of the highest draw of $z$ draws from the continuous distribution F is given by

2) $$zF^{z-1}f(x)(x)$$

What is the intuition behind this? Perhaps with some clarification of 1), I could better understand 2)?

Answer 1

The term distribution here must mean distribution function. So, if $X$ is a random variable, the distribution function is $F(x)=P(X \le x)$. When we are to find the distribution function of the maximum value obtained in $z$ independent realizations of $X$ what we really want to know is the distribution function of the random variable $max(X_1,X_2,\dots,X_z)$, that is:$$P(X_1 \le x, X_2\le x,\dots, X_z ,\le x)$$ which, by virtue of independence is $$P(X_1 \le x) \cdot P(X_2 \le x) \cdots P(X_z \le x)$$ and this equals $F(x)^z$.

The probability density function in the continuous case is obtained by differentiating the distribution function, so taking derivatives of $F(x)^z$ and using $F^\prime(x)=f(x)$ you get the stated result.

Answer 2

You asked specifically about the maximum observation $W$ among $n$ observations from $\mathsf{Unif}(0,1),$ for which the CDF is $F(x) = x,$ for $0 <x <1.$ So $F_W(x) = x^n$ and $f_W(x) = nx^{n-1},$ which is the pdf of $\mathsf{Beta}(n, 1).$ (See Wikipedia or your text.)

This is illustrated below by simulation of a million realizations with $n = 5$ in R statistical software. The histogram approximates the distribution of $W.$ The red curve is the exact density of $\mathsf{Beta}(5,1).$

n = 5;  m = 10^6;  x = runif(m*n)
MAT = matrix(x, nrow=m)            # m x n matrix, each row a sample of 5
x = apply(MAT, 1, max)             # vector of m maximums
hist(x, prob=T, col="skyblue2", 
     main="Approx Density of Max of 5 Std Uniforms")
  curve(dbeta(x, n, 1), lwd=2, col="red", add=T)