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\textbf{\Large MAT 3701:  Assignment 7 \hfill My Name}
\section{Exercises from the text}

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\section{Additional Problems}

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\item Two fair cubical dice are rolled.  Let $X$ be the value of the first and $Y$ the value of the second.  (So $X$ and $Y$ are independent random variables with density $f(x)=\frac{1}{6}$, for $x= 1, 2, 3, 4, 5, 6$.)  Let $Z=\min (X,Y)$ and $W=\max(X,Y)$.

\begin{enumerate}
\item Prove that $EZ \leq EW$ (without computing their densities).
\item Compute the densities and distribution functions of $Z$ and $W$.

\item Compute the probability that $Z=X$.
\item Compute the probability that $W=X$.
\end{enumerate}

\item ``Suppose $X$ and $Y$ are two random variables such that $P(|X-Y| \leq M)=1$ for some constant $M$.  Show that if $Y$ has finite expectation, then $X$ has finite expectation and $|EX-EY| \leq M$."  (Source:  \emph{Introduction to Probability Theory} by Hoel, Port, and Stone, 1971, p. 104.)

\item Suppose $X$ and $Y$ are independent random variables with finite second moment, $EX=2$, $EY=7$, $\text{Var} X = 1$, and $\text{Var} Y = 2$.  What is $E(2X^2 + XY + XY^2 + X^2Y^2 + 5)$?  Explain fully.

\item  
\begin{enumerate}
\item Let $X$ be a discrete random variable with uniform distribution taking the values  $-n, -n+1, -n+2, \ldots , n-2, n-1, n$.  That is, $X$ takes each of these values with equal probability $\frac{1}{2n+1}$.  Compute the mean and variance of $X$ (in terms of $n$).

\item  Let $X$ be a discrete random variable with uniform distribution taking the values $0, 1, 2, \ldots ,  2n$.  Compute the mean and variance of $X$ (in terms of $n$).

\item Let $X$ be a discrete random variable with uniform distribution taking the values \\ $-2n, -2n+2, -2n+4, \ldots , 2n-4, 2n-2, 2n$.  Compute the mean and variance of $X$ (in terms of $n$).

\end{enumerate}

\item Is it possible for $\text{E}X^2$ to be less than $(\text{E}X)^2$?  Explain.

\item  A bolt manufacturer knows that $5\%$ of her production is defective.  She gives a guarantee on a shipment of 10,000 parts by promising to refund the money if more than $a$ bolts are defective.  Based on Chebysev's inequality, how small can the manufacturer choose $a$ and still expect to give a refund no more than $1\%$ of the time?  (In MAT 3702 you will learn how to get  a much closer estimate using the normal approximation to the binomial distribution.  And, of course, you can also use tables or statistical software to calculate the correct figure with any desired accuracy directly from the binomial distribution.)  (Adapted from \emph{Introduction to Probability Theory}, by Hoel, Port, and Stone, 1971, p. 107.)

\item  The goal of this exercise is to prove that covariance is \emph{symmetric, bilinear, and non-negative definite}.  Recall that, by definition,  $Cov(X,Y) =  E(X-\mu_X)(Y-\mu_Y) = E(XY) - (EX)(EY)$.

\begin{enumerate}
\item Symmetry:  Prove that $\text{Cov}(X,Y)= \text{Cov}(Y,X)$.  (This should be obvious, from the commutativity of multiplication.)
\item Bilinearity:  Prove that $\text{Cov}(aX+bY, Z) = a\text{Cov}(X,Z) + b\text{Cov}(Y,Z)$.  Combine this result with (a) to prove linearity in the second coordinate as well:  $\text{Cov}(Z, aX+bY) = a\text{Cov}(Z,X) + b\text{Cov}(Z,Y)$.

\item Non-negative definiteness:  Prove that $Cov(X,X) \geq 0$.
\end{enumerate}

\item Let $X_1, X_2, X_3, \ldots , X_n$ be random variables having finite second moment.  (This provision ensures that the variance of each $X_i$ is well-defined and finite.)  Prove that $\text{Var} (\sum_{i=1}^n X_i) = \sum_{i=1}^n \text{Var}X_i + 2 \sum_{i=1}^{n-1} \sum_{j=i+1}^n \text{Cov} (X_i, X_j)$.


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