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Part BCSIR NET June 2023order-statistics-of-bernoulli

Order statistics of bernoulli

Let $X_{1}, X_{2}, X_{3}, X_{4}$ be i.i.d. Bernoulli(1/3) and $X_{(1)} \leq \dots \leq X_{(4)}$ the order statistics. Which of the following is true?

A. $X_{(1)}$ and $X_{(4)}$ are independent.
B.Expectation of $X_{(2)}$ is 1/2.
C.Variance of $X_{(2)}$ is 8/81.✓
D. $X_{(4)}$ is a degenerate random variable.

Solution

$X_{(2)} = 1$ iff at least three of the four are $1 : P = 4 (1/3)^{3} (2/3) + (1/3)^{4} = 9/81 = 1/9. A$ Bernoulli(1/9) has variance (1/9)(8/9) = 8/81.

The trap it tests

Execution slip

The idea was right. The computation was not.

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Related counterexample: The maximum of independent exponentials is exponential

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