Discussion
Suppose \(X_{1}, X_{2}, X_{3}\) are independent and identically distributed random variables each having exponential distribution with mean 1. Then the probability density function of sample range is given by,
Answer: D
The density of the range is \(h(r) = 2(1 - e^{-r})e^{-r}\), \(r \gt 0\).
For \(n\) independent standard exponentials, the spacings of the order statistics are independent exponentials with rates \(n - 1, n - 2, \ldots\) after the minimum. So the range \(X_{(3)} - X_{(1)}\) is the sum of exponentials with rates 2 and 1, which is the maximum of two independent standard exponentials. Its distribution function is \((1 - e^{-r})^{2}\), and differentiating gives
\[ h(r) = 2(1 - e^{-r})e^{-r}, \quad r \gt 0 \]
\(e^{-r}\) is the density of one exponential, and the other options lack the factor 2, so they do not integrate to 1, or are defined on all of \(\Re\), where a range cannot be negative.
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