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\textbf{\Large MAT 3701:  Assignment 6 \hfill My Name}
\section{Exercises from the text}

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\section{Additional Problems}


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\item  As in a previous problem, 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.

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\item Let $Y_i$ be the score obtained form the $i^{th}$ dart, $i = 1, 2, 3, 4, 5$.  Compute the density of $Y_i$.

\item Compute $\text{E}Y_i$.

\item  Let $\overrightarrow{Y} = (Y_1, Y_2, Y_3, Y_4, Y_5)$.  Compute the density of $\overrightarrow{Y} $.

\item  Let $X = \sum_{i=1}^5 Y_i$.  That is, $X$ represents the total score from the five darts.  Compute $\text{E}X$.  (Hint:  There is an easy way!)

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\item We proved in class for any random variables $X$ and $Y$ and any real constant $c$ that E$(X + Y) = \text{E}X + \text{E}Y$ and that E$cX = c\text{E}X$.

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\item Combine these results to conclude that, for any linear combination $aX + bY$ of random variables (where $a$ and $b$ are real constants), E$(aX + bY) = a\text{E} + b\text{E}Y$.

\item More generally, prove by induction that for any random variables $X_1, X_2, X_3, \ldots, X_n$ and real constants $c_1, c_2, c_3, \ldots, c_n$, E$\left(\sum_{i = 1}^n c_iX_i\right) = \sum_{i=1}^n c_i \text{E}X_i$.
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\item Suppose $X$, $Y$, and $Z$ are random variables with finite expectation, and $EX=2$, $EY=7$, and $EZ=3$.  What is $E(2X + Y + 4Z)$?  Explain.

\item ``Let $X$ be a geometrically distributed random variable [as defined by Hoel, Port, and Stone, slightly different from our text:  $f(x)=(1-p)^xp$] and let $M >0$ be an integer.  Set $Z = \min (X,M).$ Compute the mean [expectation] of $Z$."  (Source:  \emph{Introduction to Probability Theory} by Hoel, Port, and Stone, 1971, p. 104.)

\item ``Let $X$ be a geometrically distributed random variable [as defined by Hoel, Port, and Stone:  $f(x)=(1-p)^xp$] and let $M >0$ be an integer.  Set $Z = \max (X,M).$ Compute the mean  of $Z$."  (Source:  \emph{Introduction to Probability Theory} by Hoel, Port, and Stone, 1971, p. 104.)

\item  Let $X$ have hypergeometric density.  Compute $\text{E}X$.  Hint:  The easiest way to do this is as follows, and is very similar to the method we used to compute the expectation of a binomial random variable.  Let $X_i=1$ if the $i^{th}$ object picked is marked, and $0$ if it is not marked.  Thus, each $X_i$ is Bernoulli.  What is $P(X_i=1)$?  We then have $X = \sum_{i=1}^n X_i$, just as for a binomial random variable, except that the $X_i$ are not independent in this case.  

\item Let $X$ be a discrete random variable with probability distribution $f$.  Let $\mu = \text{E}X$, and let $Y = X - \mu$.  Prove the E$Y = 0$.  (Hint:  Remember that $\sum_{x}f(x) = 1$.)

\item Let $X$ be a continuous random variable with probability density $f$.  Let $\mu = \text{E}X$, and let $Y = X - \mu$.  Prove the E$Y = 0$.

\item Let $X$ be a continuous random variable with probability density $f$.  Prove that if $f$ is an even function, then E$X=0$.  (Recall that by definition a function $f$ is \emph{even} if, for all $x \in \mathbb{R}$, $f(-x) = f(x)$, and a function $f$ is \emph{odd} if, for all $x \in \mathbb{R}$, $f(-x) = -f(x)$.  Hint:  If $f$ is an even function, what type of function is the one whose output at $x$ is $xf(x)$?)

\item Does the converse to the previous problem hold?  If so, prove it;  if not, give a counterexample.

\item Let $X$ be a continuous random variable such that E$X = 0$.  Let $f$ be the probability density of $X$, and define $g$ by $g(y) = f(y-\mu)$, where $\mu \in \mathbb{R}$ is a constant.  Prove that $g$ is a probability density and that if $Y$ is a random variable with density $g$, then E$Y = \mu$.

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\subsection{Probability Review}

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\item  ``If Sam and Peter are among $n$ men who are arranged at random in a line, what is the probability that exactly $k$ men stand between them?"  (Source:  \emph{Introduction to Probability Theory} by Hoel, Port, and Stone, 1971, p. 46.)

\item ``Suppose a random sample of size $n$ is drawn from a population of $r$ objects without replacement.  Find the probability that $k$ given objects are included in the sample."   (Source:  \emph{Introduction to Probability Theory} by Hoel, Port, and Stone, 1971, p. 48.)

\item Answer the question above if the sample is drawn with replacement.

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