Discussion
Suppose X is an absolutely continuous non-negative real-valued random variable whose distribution function is \(F(x)\) and probability density function is \(f(x)\). Suppose it is given that \(f(x)/[1 - F(x)] = k\), a non-negative constant. Then which of the following statements is true ?
Answer: B
The density is \(f(x) = k\,e^{-kx},\ x \gt 0\).
\(f(x)/[1 - F(x)]\) is the hazard (failure) rate. Writing \(S = 1 - F\), we have \(f = -S'\), so
\[ \begin{aligned} -\frac{S'(x)}{S(x)} &= k \\ S(x) &= e^{-kx} \quad (S(0) = 1) \\ f(x) &= ke^{-kx} \end{aligned} \]
A constant hazard rate characterises the exponential distribution (its memoryless property). \(f(x) = 1\) on \((0, \infty)\) is not a density at all, adding a constant C would make the total probability infinite, and the step function in (D) does not integrate to 1 either.
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