Question

If $X, X_1, X_2$ are independent and identically distributed positive random variables with distribution function $F_X(x)$ then $\int_{0}^{\infty} 2 \cdot x \cdot \overline{F}_X^2(x) dx$ equals

• A. $E(\min(X_1, X_2))$
• B. $E(\max(X_1, X_2))$
• C. $E(\min^2(X_1, X_2))$
• D. $E(\max^2(X_1, X_2))$

Hint:

The survival function of the minimum of $n$ i.i.d. variables is $[\overline{F}(x)]^n$. For the maximum, the CDF is $[F(x)]^n$. Always keep the "Integral of Survival Function" method in mind for non-negative random variable expectations.

Answer 1

The Correct Option is C