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\begin{center}
\Huge \textbf{Exercise 2}
\end{center}
\vspace{1cm}

\begin{enumerate}
\item Find the distribution functions corresponding to the following density
functions:

\begin{tabular}{llll}
(i)& $f_X(x) = 1/[\pi (1 + x^2)]$& $-\infty <x<\infty $ &(Cauchy)\\
(ii)&$f_X(x) = e^{-x}/(1 + e^{-x})^2$& $-\infty <x<\infty $& (Logistic)\\
(iii)& $f_X(x) = (a-1)/(1 + x)^a$&  $0<x<\infty $& (Pareto)\\
(iv)& $f_X(x) = c\tau x^{\tau -1}e^{-cx^\tau }$&  $0<x<\infty$, $\tau >0$, $c>0$& (Weibull).\\
\end{tabular}

\item Find (without generating functions) the mean and the variance for the
following distributions

\begin{tabular}{lll}
(a)& $f_X(x) = \left\lbrace \begin{array}{ll}
                             e^{-kx}x^{r-1}k^r/(r-1)! &x\geq 0\\
                             0   &x<0
                             \end{array}\right. $& $r>0$, $k>0$\\
      &(The Gamma Distribution)\\
(b)& $f_X(x) = e^{-\lambda }\lambda ^x/x!$ & $x = 0,1,2,\dots$. $\lambda  > 0$\\
     &(Poisson Distribution)\\
(c)& $f_X(x) = \left( ^{a+x-1}_{\ \ \ x}\right) \left[
\frac{b}{1+b}\right] ^a\left[ \frac{1}{1+b}
\right] ^x$& $x = 0,1,2,\dots$. $a > 0$, $b > 0$\\
  &(Negative Binomial)\\
(d)& $f_X(x)= \frac{a-1}{(1+x)^a}$& $x > 0$, $a > 1$\\
&(Pareto)
\end{tabular}

\item Show that if a random variable $X$ is normally distributed with density,
$f(x) = \frac{1}{\sigma \sqrt{2 \pi}} e^{-(x-\mu)^2/(2 \sigma^2)}
\quad$ for $-\infty < x < \infty$, then $E(X) = \mu$ and $\var(X)
= \sigma^2$.

\item Suppose that $X$ is a continuous random variable taking values between
$-\infty $ and $\infty $ with distribution function $F(x)$.
Sometimes we want to {\it fold} the distribution of $X$ about the
value $x = a$, that is we want the distribution function $F_Y(y)$
of the random variable $Y$ obtained from $X$ by taking $Y = X - a$
if $X > a$ and $Y = a - X$ if $X < a$.   Find $F_Y(y)$ by working
out directly $P(Y \leq  y)$.   What is the density function of
$Y$? A particularly important application is the case when $X$ has
a $N(\mu ,\sigma ^2)$ distribution, and $a = \mu $.   Apply your
result to this case.

\item Show that if $X$ is a continuous positive random variable
$$
E[X] = \int  ^\infty _0[1 - F_X(x)]dx\hbox{ .}
$$
(Try integration by parts.)

\item If $X$ is a continuous positive continuous variable with
density function $f_X(x)$ and mean $\mu$, show that
$$
f(y) = \left\{\begin{array}{ll}
              0& y < 0\\
              yf_X(y)/\mu& y \ge 0
              \end{array} \right.
$$
is a density function, and hence show that
$$
\mbox{E}(X^3)\mbox{E}(X) \ge \{\mbox{E}(X^2)\}^2.
$$
\newpage

\item A random variable has `no memory' if for all $x$ and for $y > 0$
$$
P[X > x+y \mid X > x] = P[X > y].
$$
Show that if $X$ has either the exponential distribution, or a
geometric distribution with Prob$(X = x) = q^{x-1}p$, then $X$ has
no memory.  Interpret this property.

\item In a simple queueing system, the time spent in the queue has
the following distribution function
$$
F(x) = \left\lbrace \begin{array}{lll}
                         0& x < 0\\
                         1-\rho \exp(-\lambda (1-\rho )x)& x \geq  0
                     \end{array}\right.
$$
where $\lambda \ge 0$ and $0<\rho < 1$.
 Check that this is a distribution function, and describe the
form of the distribution in words.

\item If $X$ is a nonnegative continuous random variable, then
$$
\bar{F}(x) = P(X > x)
$$
is called the {\em survival function}, and
$$
f(x)/\bar{F}(x)
$$
is called the {\em failure rate} at $x$.
\begin{enumerate}
\item Find the failure rate for the Weibull and Pareto distributions.
\item Show that in general the failure rate does not decrease as $x$ increases
if
$$
\bar{F}(x + y)/ \bar{F}(x)
$$
does not increase as $x$ increases, for all $y \geq  0$.
(Differentiate the logarithm of the given expression with respect
to $x$.)
\end{enumerate}

\end{enumerate}
