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\lecture{CS5350: Machine Learning}{HW8: Probabilistic Modeling}{Due 18 Nov 2008}

\section{Written Exercises}

\bee
\i Suppose we have a binomial distribution with the ``probability of
heads'' $\pi = 0.8$.  Compute (show all the stops) the expected value
and variance of this distribution.

\i Suppose we have a Gaussian with known mean $\mu = 1$ and known
variance $\si^2 = 1$.  What is the \emph{density} of the distribution
$\Nor(\mu, \si^2)$ at the following points: $0, 1, 2$?

\i Consider the previous question, but where $\si^2 = 0.1$.  What is
the density at the given points?  For $x=1$, the density should be
\emph{greater than one}.  How is this possible given that the Gaussian
is normalized (i.e., sums to one).

\i The \emph{Poisson} distribution is a distribution over
\emph{positive count values}.  It has the form $p(k \| \la) = \frac 1
{e^{\la}} \frac {\la^k} {k!}$, where $k$ is the count and $\la$ is the
(single) parameter of the Poisson.  Suppose we have a bunch of count
data (for instance, the number of cars to pass an intersection on a
given day, measured on $N$-many days) called $k_1, k_2, \dots, k_N$.
Compute the maximum likelihood estimate for $\la$ given this data.
(Hint: write down the likelihood, then take the log.  Do some algebra
to simplify and then take the derivative with respect to $\la$.)

\ene


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