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\title{Stochastic Processes}
\author{Summer Semester 2026, Exercise 3}
\date{Due Thursday, May 21, 2026}
\maketitle


\begin{enumerate}


\item Compute mean and variance of a random variable $X$ that is
Poisson-distributed with rate $\lambda$.

\emph{Hint:} Recall from class that the moment generating function of
the Poisson distribution is
\[
  M_X(t) = \e^{\lambda (\e^t-1)} \,.
\]

\item IT support receives calls at an average rate of $2$ per hour.
\begin{enumerate}
\item Assuming that calls are Poisson-distributed, compute the
probability that one call is received in two hours.
\item Likewise, compute the probability that two or more calls are
received in half an hour.
\item Discuss why or why not the Poisson distribution may be an
appropriate model for the distribution of IT support calls.
\end{enumerate}

\item (HJ Exercise 5.43.) Let $X=(X_1, X_2, X_3)$ be a multivariate
random variable taking values in $\mathbb{R}^3$ with PDF defined for
$\boldsymbol{x} = (x_1, x_2, x_3)$ as
\[
  f_X(\boldsymbol{x}) =
  \begin{cases}
    \lVert \boldsymbol{x} \rVert^2 & \text{if } x_i \in [0,1] \text{
    for } i=1,2,3 \\
    0 & \text{otherwise} \,.
  \end{cases}
\]
\begin{enumerate}
\item Verify that
\[
  \int_{\mathbb{R}^3} f_X(\boldsymbol{x}) \, \mathrm{d} \boldsymbol{x}
  = 1 \,.
\]
\item Find $P(X_1 \leq \frac15, X_2 \leq \frac13, X_3 \leq \frac12)$.
\item Find $P(X_1 \geq \frac15, X_2 \leq \frac13, X_3 \leq \frac12)$.
\item Find $\mathbb E[X]$.
\item Find the marginal PDF $f_1(x)$ of $X_1$.
\item Find the covariance
\[
  \operatorname{Cov} (X_i, X_j)
  = \mathbb E[X_i X_j] - \mathbb E[X_i] \, \mathbb E[X_j] \,.
\]
\end{enumerate}

\item Show that you can obtain an estimate of the mathematical
constant $\pi$ by flipping a fair coin a large number of times in the
following way.
\begin{enumerate}
\item Let $X_i$ be the Bernoulli random variable of the $i$-th coin
flip.  Argue that
\[
  Z_n = 2 \, \frac{X_1 + \dots + X_n}{\sqrt n} - \sqrt n
\]
converges in distribution to the standard normal distribution.

\emph{Hint:} Use the central limit theorem.

\item Suppose $Z$ is normally distributed with mean zero and variance
one.  Show that the \emph{mean absolute deviation} (MAD) is given by
\[
  \mathbb E[\lvert Z \rvert] = \sqrt{\frac2\pi} \,,
\]
so that
\[
  \pi = \frac2{\mathbb E[\lvert Z \rvert]^2} \,.
\]
\item Use this last expression to devise a procedure for estimating
$\pi$.  Write a short Python code that simulates this experiment.

\end{enumerate}


\end{enumerate}
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