Problem source

The lifetime of a fly (measured in hours) is given by the continuous random variable $T$ with probability density function $f(t)$ and cumulative distribution function $F(t)$.  The \textit{hazard function}, $h(t)$, is defined, for $F(t) < 1$, by
  \[
  h(t) = \frac{f(t)}{1-F(t)}\,.
  \]
  \begin{questionparts}
\item Given that the fly lives to at least  time $t$, show that the  probability of its dying within the following $\delta t$ is approximately $h (t) \, \delta t$ for small values of $\delta t$. 
  \item Find the hazard function in the case $F(t) = t/a$    for $0<    t <   a$. 
   Sketch $f(t)$ and $h(t)$ in this case.
  \item The random variable $T$ is distributed on the interval $t >      a$, where $a>0$, and its hazard function is $t^{-1}$.  Determine    the probability density function for $T$.
  \item Show that $h(t)$ is constant for $t > b$ and zero otherwise if and only if $f(t) =ke^{-k(t-b)}$ for $t > b$, where $k$ is a positive constant.
  \item The random variable $T$ is distributed on the interval $t >   0$ and its hazard function is given by
    \[
    h(t) =
    \left(\frac{\lambda}{\theta^\lambda}\right)t^{\lambda-1}\,,
    \]
    where $\lambda$ and $\theta$ are positive constants.  Find the probability density function for $T$.
  \end{questionparts}

Solution source

\begin{questionparts}
\item $\,$ \begin{align*}
&& \mathbb{P}(T > t + \delta t | T > t) &= \frac{\mathbb{P}(T < t + \delta t)}{\mathbb{P}(T > t )} \\
&&&= \frac{\int_t^{t+\delta t} f(s) \d s}{1-F(t)} \\
&&&\approx \frac{f(t)\delta t}{1-F(t)} \\
&&&= h(t) \delta t
\end{align*}

\item If $F(t) = t/a$ then $f(t) = 1/a$ and $h(t) = \frac{1/a}{1-t/a} = \frac{1}{a-t}$


\begin{center}
    \begin{tikzpicture}
    \def\functionf(#1){1/1};
    \def\functiong(#1){1/(1-(#1))};
    \def\xl{-0.5}; 
    \def\xu{2};
    \def\yl{-0.5}; 
    \def\yu{5};
    
    % Calculate scaling factors to make the plot square
    \pgfmathsetmacro{\xrange}{\xu-\xl}
    \pgfmathsetmacro{\yrange}{\yu-\yl}
    \pgfmathsetmacro{\xscale}{10/\xrange}
    \pgfmathsetmacro{\yscale}{10/\yrange}
    
    % Define the reusable styles to keep code clean
    \tikzset{
        x=\xscale cm, y=\yscale cm,
        axis/.style={thick, draw=black!80, -{Stealth[scale=1.2]}},
        grid/.style={thin, dashed, gray!30},
        curveA/.style={very thick, color=cyan!70!black, smooth},
        curveB/.style={very thick, color=orange!90!black, smooth},
        curveC/.style={very thick, color=red!90!black, smooth},
        dot/.style={circle, fill=black, inner sep=1.2pt},
        labelbox/.style={fill=white, inner sep=2pt, rounded corners=2pt} % Protects text from lines
    }
    
    % Draw background grid
    \draw[grid] (\xl,\yl) grid[step=.2] (\xu,\yu);
    
    % Set up axes
    \draw[axis] (\xl,0) -- (\xu,0) node[right, black] {$x$};
    \draw[axis] (0,\yl) -- (0,\yu) node[above, black] {$y$};
    
    % Define the bounding region with clip
    \begin{scope}
        \clip (\xl,\yl) rectangle (\xu,\yu);
        \filldraw (0,0) circle (1.5pt) node[below left] {$O$};
        \draw[curveA, domain=0:1, samples=150] 
            plot ({\x},{\functionf(\x)});
        \draw[curveB, domain=0:.99, samples=150] 
            plot ({\x},{\functiong(\x)});
    \end{scope}
    

    \end{tikzpicture}
\end{center}


\item $\,$ \begin{align*}
&& \frac{F'}{1-F} &= \frac{1}{t} \\
\Rightarrow && -\ln (1-F) &= \ln t + C\\
\Rightarrow && 1-F &= \frac{A}{t} \\
&& F &= 1 - \frac{A}{t} \\
F(a)  = 0: && F &= 1 - \frac{a}{t} \\
&& f(t) &= \frac{a}{t^2}
\end{align*}

\item ($\Rightarrow$) \begin{align*}
&& \frac{F'}{1-F} &= k \\
\Rightarrow && -\ln(1-F) &= kt+C \\
\Rightarrow && 1-F &= Ae^{-kt} \\
F(b) = 0: && 1 &= Ae^{-kb} \\
\Rightarrow && 1-F &= e^{-k(t-b)}\\
\Rightarrow && f &= ke^{-k(t-b)} \\
\end{align*}

($\Leftarrow$) $f(t) = ke^{-k(t-b)} \Rightarrow F(t) = 1-e^{-k(t-b)}$ and the result is clear.

\item $\,$ \begin{align*}
&& \frac{F'}{1-F} &= \left ( \frac{\lambda}{\theta^{\lambda}} \right) t^{\lambda-1} \\
\Rightarrow && -\ln(1-F) &= \left ( \frac{t}{\theta} \right)^{\lambda} +C\\
\Rightarrow && F &= 1-A\exp \left (- \left ( \frac{t}{\theta} \right)^{\lambda} \right) \\
 F(0) = 0: && 0 &= 1-A \\
\Rightarrow && F &= 1 - \exp \left (- \left ( \frac{t}{\theta} \right)^{\lambda} \right) \\
\Rightarrow && f &= \lambda t^{\lambda -1} \theta^{-\lambda} \exp \left (- \left ( \frac{t}{\theta} \right)^{\lambda} \right)
\end{align*}

\end{questionparts}