Let \( X_1, \dots, X_n \) be independent and identically distributed \( \mathrm{Uniform}(0,1) \) random variables.

We define the maximum and minimum as:

\[ Z_n = \max{X_1, \dots, X_n}, \quad Y_n = \min{X_1, \dots, X_n} \]

Part 1: Maximum \( Z_n \)

CDF of \( Z_n \)

\[ F_{Z_n}(z) = \Pr(Z_n \le z) = \Pr(X_1 \le z, \dots, X_n \le z) = \Pr(X_1 \le z)^n = z^n, \quad 0 \le z \le 1 \]

PDF of \( Z_n \)

\[ f_{Z_n}(z) = \frac{d}{dz}F_{Z_n}(z) = n z^{n-1}, \quad 0 \le z \le 1 \]

Expected Value of \( Z_n \)

\[ E[Z_n] = \int_0^1 z \cdot f_{Z_n}(z)\,dz = \int_0^1 z \cdot n z^{n-1}\,dz = n \int_0^1 z^n\,dz = \frac{n}{n+1} \]

Part 2: Minimum \( Y_n \)

CDF of \( Y_n \)

\[ F_{Y_n}(y) = \Pr(Y_n \le y) = 1 - \Pr(X_1 > y, \dots, X_n > y) = 1 - (1 - y)^n, \quad 0 \le y \le 1 \]

PDF of \( Y_n \)

\[ f_{Y_n}(y) = \frac{d}{dy}F_{Y_n}(y) = n(1 - y)^{n-1}, \quad 0 \le y \le 1 \]

Expected Value of \( Y_n \)

\[ E[Y_n] = \int_0^1 y \cdot f_{Y_n}(y)\,dy = n \int_0^1 y(1 - y)^{n-1}\,dy = \frac{1}{n+1} \]

(by the Beta integral identity)

Final Answers

For \( Z_n = \max(X_1,\dots,X_n) \):

  • CDF: \( F_{Z_n}(z) = z^n \)
  • PDF: \( f_{Z_n}(z) = n z^{n-1} \)
  • Expected value: \( \boxed{E[Z_n] = \frac{n}{n+1}} \)

For \( Y_n = \min(X_1,\dots,X_n) \):

  • CDF: \( F_{Y_n}(y) = 1 - (1 - y)^n \)
  • PDF: \( f_{Y_n}(y) = n(1 - y)^{n-1} \)
  • Expected value: \( \boxed{E[Y_n] = \frac{1}{n+1}} \)

Reference