%source: fall99 final exam problem 8
%topic: Markov Theorem

\begin{problem}[10]

% COMMENTS: Seems fine!

A herd of 350 cows is stricken by an outbreak of "mad cow" disease.  The
disease lowers the normal body temperature of a cow, and a cow will die if
its temperature goes below 90 degrees F.  The disease epidemic is so
intense that it lowered the average temperature of the herd to $45$
degrees.  Body temperatures as low as $20$ degrees, but no lower, were
actually found in the herd.

\bparts

\ppart[5] State a best possible upper bound on the number of cows
whose temperature is high enough to survive.

\solution[\vspace{4cm}]{

Let $R$ be the temperature.  Apply Markov's Bound to $R-20$:
\[
\prob{R \geq 90} = \prob{R-20 \geq 70} \leq \expect{R-20}/70 = (45-20)/70 = 5/14,
\]
so at most $(5/14)350 = \textbf{125}$ cows have temperature $\geq 90$
degrees.

}

%\vspace{3in}

\ppart[5] Give an example set of temperatures that achieve your bound.

\solution[\vspace{4cm}]{

Of the 350 cows, 125 have 90 degree temperature and 225 have 20
degree temperature.  The mean is
\[
(125 \cdot 90 + 225 \cdot 20)/350 = 45,
\] as required.
}

%\vspace{2in}

\eparts

\end{problem}
