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\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 335 \\
    Laboratory \#1, Rev. A, due Monday, 2024-09-09 }
\end{center}

\vspace{.3 in}

The purpose of this lab is to {\it theoretically} and {\it experimentally} quantify the distribution of the random variable $X$ representing {\bf Monopoly rolls}: the sum of two thrown dice.  Perform and analyze the experiment in pairs, but submit your own separate assignment with all plots and calculations.

{\bf 1.} Theoretical estimation of the {\bf Parent Distribution}
 
{\bf \quad a)} Identify the {\it sample space} of possible outcomes labeled by $x=x_{1}+x_{2}$, where $x_{i}$ is the number of each die, and the {\it probability} $P(x)$ for each individual outcome (the {\it events}).

{\bf \quad b)} What assumptions went into this model?

{\bf \quad c)} Draw step plot of the {\it distribution} $P(x)$, labeling your axes in the figure, spanning the entire width of your sheet.

{\bf \quad d)} Calculate the mean $\mu$, median $\mu_{1/2}$, mode $\mu_{max}$,  standard deviation $\sigma$, and average deviation $\alpha$ of this distribution.

{\bf \quad e)} Draw the {\it central values} as ticks and {\it deviations} as error bars on the abscissa of your plot, on separate lines with $(\mu,\sigma)$ and $(\mu_{1/2},\alpha)$ together and $\mu_{max}$ alone.

{\bf 2.} Experimental measurement of a {\bf Sample Distribution}

{\bf \quad a)} Perform an experiment to estimate $P(x)$ by throwing a pair of dice $N=25$ times and recording each sum $x_{i}$ for $i=1\ldots20$.

{\bf \quad b)} Histogram the number of occurrences of each value of $x$ in a {\it frequency table}.

{\bf \quad c)} Plot the the {\it sample distribution} $P_{M}(x)$ of $\{x_{i}\}$, normalized (divide by $N$) to approximate the {\it parent distribution} $P(x)$  Plot this distribution as a disconnected solid dot for each value of $x$.

{\bf \quad d)} Recalculate the sample mean $\bar x$, median $x_{1/2}$, mode $x_{max}$, standard deviation $s$, and average deviation $a$ of your sample distribution, dividing by $N-1$ instead of $N$ for the deviations.

{\bf \quad e)} Plot the {\it central values} and {\it deviations} as before with a solid dot on the tick.


{\bf 3.} Calculation of the {\bf Combined Distribution}

{\bf \quad a)} Add up the histograms from each group to obtain the combined sample distribution $P_{C}(x)$, and plot it on the same graph with open squares.

{\bf \quad b)} Recalculate and plot the same {\it central values} and {\it deviations} as before with an open square on the tick.

{\bf \quad c)} Qualitatively compare the plots and statistics of the parent, sample, and combined distributions.  [{\it bonus: How do the error bars relate to the difference between the different ticks?}]

\newpage

{\bf 4.} In the game {\bf Monopoly}, each player rolls two dice to advance their piece around the board.

{\bf \quad a)} How often does a player leave jail and end on free parking without passing it (ignoring the effects of cards, and rolling again on doubles)?

{\bf \quad b)} In light of your analysis please discuss this heat map of the number of landings on each space of a Monopoly board.  Why aren't all spaces visited equally?

\includegraphics{X01_fig1}

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