Discussion
Which of the following statements about an exponential distribution with mean \(\lambda\) is true ?
Answer: D
The true statement is the memoryless property \(\bar{F}(t + s) = \bar{F}(t)\bar{F}(s)\).
With mean \(\lambda\), the survival function is \(\bar{F}(x) = e^{-x/\lambda}\), so \(\bar{F}(t + s) = e^{-(t+s)/\lambda} = \bar{F}(t)\bar{F}(s)\) for all \(t, s \ge 0\). The exponential is the only continuous distribution with this property.
The others are wrong. The variance is \(\lambda^{2}\), not \(\lambda\). The m.g.f. \(1/(1 - \lambda t)\) exists only for \(t \lt 1/\lambda\), not for all real \(t\). And \(\int_{0}^{\infty} e^{-\lambda x}\, dx = 1/\lambda\), which is the mean only for the rate parametrisation; here the mean is \(\lambda = \int_{0}^{\infty} e^{-x/\lambda}\, dx\).
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