Discussion
Suppose \(\{X_{n}, n \ge 1\}\) is a sequence of independent and identically distributed random variables each following uniform U(2, 6) distribution. Which of the following is not true ?
Answer: C
The false statement is that \(X_{(n)}\) converges in law to an exponential distribution.
For U(2, 6), \(P(X_{(n)} \le 6 - \epsilon) = \left(\frac{4 - \epsilon}{4}\right)^{n} \to 0\), so the maximum converges in probability to 6, and hence in law to the degenerate distribution at 6, not to an exponential. Likewise \(X_{(1)} \to 2\) in probability, and the sample median converges in probability to the population median 4.
An exponential limit appears only after rescaling: \(n(6 - X_{(n)})\) converges in law to an exponential distribution with mean 4. Without the factor \(n\) the limit is a point mass. So the first, second and fourth statements are true and the third is the one that is not.
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