91Ó°ÊÓ

A random variable X for which P(X>0)=1 and the PDF is f and the CDF is F. We define a function h as follows

\(h\left( x \right) = \frac{{f\left( x \right)}}{{1 - F\left( x \right)}}\,\,,x > 0\)

We need to show that h(x) is constant ifX has exponential distribution.

X has exponential distribution, using memoryless property of exponential distribution we get

\(\begin{aligned}{c}P\left( {X \ge x} \right) &= {e^{ - \lambda x}}\,,\lambda > 0\\1 - P\left( {X \le x} \right) &= {e^{ - \lambda x}}\\1 - F\left( x \right) &= {e^{ - \lambda x}}\end{aligned}\)

Now,

\(\begin{aligned}{l}h\left( x \right) &= \frac{{f\left( x \right)}}{{1 - F\left( x \right)}}\\\,\,\,\,\,\,\,\,\,\,\,& = \frac{{\lambda {e^{ - \lambda x}}}}{{{e^{ - \lambda x}}}}\\\,\,\,\,\,\,\,\,\,\,\,& = \lambda \end{aligned}\)

Hence proved that h(x) is constant if X has exponential distribution.