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\textbf{\Large MAT 3701:  Assignment 8 \hfill My Name}

\section{Exercises from the Text}

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\section{Additional Exercises}

Recall that the \emph{correlation coefficient} of two random variables, $X$ and $Y$, is defined to be $$\rho(X,Y) = \frac{Cov(X,Y)}{\sigma_X \sigma_Y}.$$

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\item  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 marked objects in the sample.  (As we know, such a random variable is said to have \emph{hypergeometric density} with the appropriate parameters.)

Compute $\text{Var} X$.  Hints:  As in computing $\text{E}X$ on the previous problem set, let $X_i$ be the Bernoulli random variable that records whether or not a marked object is obtained in the $i^{th}$ drawing, and use the formula from the previous problem.  To compute $\text{Cov} (X_i, X_j) = \text{E}X_i X_j - \text{E}X_i \text{E}X_j$, note that $X_i X_j = 0$ except when $X_i = X_j = 1$.

\item Suppose the random vector $(X,Y)$ takes the values $(1,0), (0,1), (-1,0)$, and $(0,-1)$ with equal probability.  Prove that $X$ and $Y$ are \emph{not} independent, but nonetheless $\rho(X,Y) = 0$.

\item  A box has 3 red balls and 2 black balls.  A random sample of size 2 is drawn without replacement.  Let $U$ be the number of red balls selected and let $V$ be the number of black balls selected.  Compute $\rho(U,V)$.

\item Suppose a box has 3 balls labeled 1, 2, and 3.  Two balls are selected without replacement from the box.  Let $X$ be the number on the first ball and $Y$ be the number on the second ball.  Compute $\text{Cov}(X,Y)$ and $\rho(X,Y)$.

\item A doctor's office receives, on average, 10  calls during the lunch hour each day.  Of course, the number of calls varies from day to day, and the office wants to make sure it has adequate staff to answer the maximum number of calls they are likely to get.  To that end, the office would like to know the answer to the following question:  On any given day, what is the probability that the office will receive more than 15 calls during the lunch hour?  (This is too hard to calculate by hand;  use Table II and a calculator!  Alternatively, you can probably find a good Web resource.  Give your answer to the nearest whole percentage.)

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\section{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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