Discussion
Which of the following is not true for an exponential distribution ?
Answer: C
The false statement is the distribution of the nth order statistic is exponential.
For n independent exponential variables with rate \(\lambda\), the largest one has
\[ P(X_{(n)} \le x) = (1 - e^{-\lambda x})^{n} \]
which is not an exponential distribution function for \(n \ge 2\).
The other statements are true. The exponential is memoryless, \(P(X \gt s + t \mid X \gt s) = P(X \gt t)\); its hazard rate \(f(x)/(1 - F(x)) = \lambda\) is constant; and the minimum satisfies \(P(X_{(1)} \gt x) = e^{-n\lambda x}\), so it is exponential with rate \(n\lambda\).
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