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\textbf{\Large MAT 3701:  Assignment 4 \hfill My Name}
\section{Exercises from the text}

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\section{Additional Problems}

\subsection{Probability Spaces}
\emph{The following problems are based on the axioms of a probability space.  All proofs must be logical, rigorous, and complete!}

\begin{enumerate}

\item Given a measure on $\mathbb{R}$, prove that every open interval is an event (that is, measurable) if and only if every closed interval is an event.  (Hint:  Use the closure of a $\sigma$-algebra under countable unions and intersections.)  Note that this result extends immediately to any measure space $\mathcal{M} \in \mathbb{R}$.  We will be concerned with probability spaces, of course, but the above result does not depend on the measure of the entire space being $1$.


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In the problems that follow, let $\rho$ be a probability measure on the positive reals, $(0,\infty)$, and define the function $F:  (0,\infty) \rightarrow [0,\infty)$ by $F(t) = \rho \big( (0,t])$.  (You should recognize that if the random variable $T$ is defined by the outcomes of this sample space, then $F$ is the cumulative distribution function of $T$.)

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\item  Prove that $F$ must be an increasing function on $(0,\infty)$.  (Recall that a function $f:  (a,b) \rightarrow \mathbb{R}$ is \emph{increasing} if, for all $x_1, x_2 \in (a,b)$, $x_1 < x_2 \Rightarrow f(x_1) \leq f(x_2)$.  The function $f$ is \emph{strictly} increasing if the inequality between the outputs is strict.  Note that the function $F$ of this exercise need not be strictly increasing.)

\item Prove that $F$ must be right continuous on $(0,\infty)$;  that is, $\lim_{x \rightarrow t^+} F(x)= F(t)$.  (Hint:  Use the result on intersections of nested events from the previous problem set.)

\item Prove that if $F$ is continuous function, then $\rho \big( (0,t)) = F(t)$ as well.  (Hint:  Use the result on unions of nested events from the previous problem set.)  Conclude that $\rho \big(\{t\}) = 0$, for any $t>0$.  (Note that condition that $F$ be continuous is not always satisfied;  there are probability measures such that $F(t) = \rho \big((0,t] \big)  > \rho \big((0,t))$  In this case, $\rho (\{t\}) > 0$.)

\item Prove the converse of the result above:  if $\rho \big(\{t\}) = 0$ for all $t \in (0,\infty)$, then $F$ is continuous.

\item Prove that the probability an atom decays at any particular instant must be 0;  conclude that for the measure giving the probability the atom decays during any interval of time, $F$ as defined above is continuous.  Why does this not cause a contradiction to the probability of the entire sample space, $(0,\infty)$, being 1?


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\subsection{Random Variables and Vectors}

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\item  Prove that if $X$ is a discrete random variable, then $\{x \in \mathbb{R}: P(X=x) > 0 \} \neq \emptyset$.  (Hint:  Use proof by contradiction.)

\item  Suppose a fair coin is tossed $5$ times.  Define the random variable $X$ to be the number of heads obtained.  What is the probability that $X=x$, for $x = 0, 1, 2, 3, 4, \text{ or } 5$?  (This gives a formula for the probability density (distribution) of $X$.)

\item A fair die with six faces is tossed 5 times.  Let $X$ be the number of sixes that occur.  Give a formula for the probability density (distribution) of $X$.

\item A fair coin is tossed until a head is obtained.  Let $X$ denote the number of the toss on which this head occurs.  

\begin{enumerate}

\item Give a formula for the probability density (distribution) of $X$.

\item What is the probability that the first head is obtained on either the second or third toss (that is, $X = 2$ or $X=3$)?

\item What is the probability that $X$ is even?

\item What is the probability that $X$ is odd?

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\item A fair six-sided die is tossed until a six is obtained.  Let $X$ be the number of the toss on which this six occurs. 

\begin{enumerate}

\item Give a formula for the probability density (distribution) of $X$.

\item What is the probability that $X$ is even?

\item What is the probability that $X$ is odd?

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\item  Two fair six-sided dice are tossed.  Let $X$ be their sum.  Give a formula for the probability density (distribution) of $X$.

\item A fair die with six faces is rolled five times.

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\item What is the probability that one of each number from one to five is rolled (but no six).

\item What is the probability that the five rolls are all different?

\item What is the probability that two threes are rolled first, followed by three ones?

\item What is the probability that two threes and three ones are rolled (in any order)? 

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\item A fair die with six faces is rolled seven times.

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\item What is the probability that two threes are rolled first, followed by one six, and then four ones?

\item What is the probability that two threes, one six, and four ones are rolled (in any order)?

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\item  A fair cubical die is tossed seven times.  Let $X_1$ be the number of ``ones" obtained among the tosses, let $X_2$ be the number of ``twos" obtained, and so on.  Let $\overrightarrow{X} = (X_1, X_2, X_3, X_4, X_5, X_6)$.

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\item Compute the probability density (distribution) of $\overrightarrow{X}$ (which is generally called the \emph{joint probability density (distribution)} of $X_1, X_2, X_3, X_4, X_5, X_6$).

\item Compute the density of $X_i$ for each $i = 1, 2, 3, \ldots, 7$.  ($X_i$ is generally called the \emph{$i^{th}$ marginal probability density (distribution)} of $\overrightarrow{X}$.)

\item Let $f$ denote the probability density (distribution) of $\overrightarrow{X}$, and let $f_i$ denote the $i^\text{th}$ marginal probability density (distribution) of $\overrightarrow{X}$.  Show that $f_i(x) = \sum_{x_i = x} f(x_1, x_2, x_3, \ldots, x_6)$.

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\item  A target is made by drawing concentric circles of radius 1, 2, 3, and 4 meters.  Five darts are thrown at the target ``at random";  that is, the darts are guaranteed to hit the target, and the probability a dart will land in any region is proportional to its area.  Each throw is independent of the others. 

The ``bull's eye" within the circle of radius 1 is assigned 4 points.  The annular region between the circle of radius 1 and the circle of radius 2 is assigned 3 points.  The annular region between the circle of radius 2 and the circle of radius 3 is assigned 2 points.  The outermost annular region is assigned 1 point.

Let $X$ be the score obtained from the five darts.

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\item What are the possible values of $X$?

\item Compute the probability density of $X$.

\item Compute the (cumulative) distribution function of $X$.

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\item  There are 10 balls in an urn.  Of these, 3 are red and the rest are black.  Five balls are drawn at random from the urn, without replacement.  Let $X$ be the number of red balls among those chosen.

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\item What are the possible values of $X$?

\item Compute the probability density of $X$.

\item Compute the (cumulative) distribution function of $X$.

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\item  More generally, let $l, m$, and $n$ be positive integers with $l \leq n$ and $m \leq n$.  Suppose there is a population of $n$ objects, and $l$ of these objects are of a marked type.  A sample of size $m$ is drawn from the objects \emph{without} replacement.  Let $X$ be the number of marked objects in the sample.  

As examples, we might have $n$ balls in an urn, of which $l$ are red and the rest are black, as in the problem above.  A sample of $m$ balls is drawn from the urn.  $X$ is the number of red balls in the sample.   In a more realistic application, there is a population of $n$ deer in a forest, of which $l$ have been marked with a tag in an ear.  Hunters shoot $m$ deer.  $X$ is the number of tagged deer that are shot.  (This number can be used to estimate the total population, as we will see later on.)

Compute the density function of $X$ (in terms of $l$, $m$, and $n$).  A random variable with this density is said to be \emph{hypergeometrically distributed}.  Compare the hypergeometric distribution with the binomial distribution: if a sample of size $m$ is drawn \emph{with replacement} from a population of size $n$ that contains $l$ marked objets, then the number of marked objects obtained is binomially distributed with parameters $m$ and $\frac{l}{n}$.  The hypergeometric density can be written in two convenient forms, one entirely in terms of combinations, and the other more analogous to the formula for the binomial density.  Give both forms in your answer.

\item A point is picked at random from a disk of radius 1.  Let $R$ be the distance of the point from the center of the target.

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\item Compute the (cumulative) distribution function of $R$.

\item Give a probability density for $R$.

\end{enumerate}

\end{enumerate}

\end{document}
 
