= Solution
For $t\ge0$ 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. Thus
$$
\boxed{f(t)=\theta e^{-\theta t},\quad S(t)=e^{-\theta t},\quad h(t)=f(t)/S(t)=\theta,\quad H(t)=\int_0^t h(u)\,du=\theta t=-\log S(t).}
$$
The <probability density function> is zero for $t<0$, and the <survivor function> is one there.
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