Jones puts in two new lightbulbs: a 60 watt bulb and a 100 watt bulb. It is claimed that the lifetime of the 60 watt bulb has an exponential density with average lifetime 200 hours ([math]\lambda = 1/200[/math]). The 100 watt bulb also has an exponential density but with average lifetime of only 100 hours ([math]\lambda = 1/100[/math]). Jones wonders what is the probability that the 100 watt bulb will outlast the 60 watt bulb.

If [math]X[/math] and [math]Y[/math] are two independent random variables with exponential densities [math]f(x) = \lambda e^{-\lambda x}[/math] and [math]g(x) = \mu e^{-\mu x}[/math], respectively, then the probability that [math]X[/math] is less than [math]Y[/math] is given by

[[math]] P(X \lt Y) = \int_0^\infty f(x)(1 - G(x))\,dx, [[/math]]

where [math]G(x)[/math] is the cumulative distribution function for [math]g(x)[/math]. Explain why this is the case. Use this to show that [math] P(X \lt Y) = \frac \lambda{\lambda + \mu} [/math] and to answer Jones's question.