Question

Let $X_1, X_2, X_3, X_4$ be i.i.d. random variables having a continuous distribution. Then $P(X_3 < X_2 < \max(X_1, X_4))$ equals

• A. $\dfrac{1}{2}$
• B. $\dfrac{1}{3}$
• C. $\dfrac{1}{4}$
• D. $\dfrac{1}{6}$

Hint:

For problems involving i.i.d. continuous random variables, use symmetry arguments since all orderings are equally probable.

Answer 1

The Correct Option is C

Step 1: Independence and symmetry.
For i.i.d. continuous random variables, all $4! = 24$ possible orderings are equally likely.

Step 2: Focus on ordering.
We need cases where $X_3 < X_2$ and $X_2 < \max(X_1, X_4)$. The probability that $X_3 < X_2$ is $\dfrac{1}{2}$. Given this, the probability that $X_2$ is not the largest among $\{X_1, X_2, X_4\}$ is $\dfrac{2}{3}$.

Step 3: Multiply probabilities.
\[ P = \frac{1}{2} \times \frac{2}{3} = \frac{1}{3}. \]

Step 4: Conclusion.
\[ \boxed{P = \frac{1}{3}}. \]