Question:
Let \( X_1, X_2 \) and \( X_3 \) be i.i.d. \( U(0,1) \) random variables. Then \( E\left( \dfrac{X_1 + X_2}{X_1 + X_2 + X_3} \right) \) equals
Let \( X_1, X_2 \) and \( X_3 \) be i.i.d. \( U(0,1) \) random variables. Then \( E\left( \dfrac{X_1 + X_2}{X_1 + X_2 + X_3} \right) \) equals
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In cases involving i.i.d. random variables, you can simplify the expected value by symmetry and the linearity of expectation.
Updated On: Dec 17, 2025
- \( \frac{1}{3} \)
- \( \frac{1}{2} \)
- \( \frac{2}{3} \)
- \( \frac{3}{4} \)
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The Correct Option is C
Solution and Explanation
Step 1: Understand the problem.
The random variables \( X_1, X_2, X_3 \) are independent and uniformly distributed over \( [0,1] \). The expected value of the ratio can be calculated using the properties of expectations. Since the random variables are identically distributed, the expected value simplifies as: \[ E\left( \frac{X_1 + X_2}{X_1 + X_2 + X_3} \right) = \frac{2}{3}. \]
Step 2: Conclusion.
The correct answer is (C) \( \frac{2}{3} \).
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