[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Assume that a new light bulb will burn out after [math]t[/math] hours, where [math]t[/math] is chosen from [math][0,\infty)[/math] with an exponential density

[[math]] f(t) = \lambda e^{-\lambda t}\ . [[/math]]

In this context, [math]\lambda[/math] is often called the failure rate of the bulb.

• Assume that [math]\lambda = 0.01[/math], and find the probability that the bulb will not burn out before [math]T[/math] hours. This probability is often called the reliability of the bulb.
• For what [math]T[/math] is the reliability of the bulb [math] = 1/2[/math]?