 Solution

For $t\ge 0$ and $\theta >0$, the exponential distribution has probability density function $f(t)=\theta e^{-\theta t}$. Integrating the probability density function gives the survivor function $S(t)=P(T>t)=e^{-\theta t}$. The hazard function divides instantaneous failure density by the probability of having survived, and the integrated hazard accumulates this rate. ThusThe probability density function is zero for $t<0$, and the survivor function is one there.