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\textbf{\Large MAT 3701:  Assignment 5 \hfill My Name}
\section{Exercises from the text}

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\section{Additional Problems}

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\item  ``Let $X$ and $Y$ be discrete random variables, and let $g$ and $h$ be functions such that the following identity holds:  $P(X=x$ and $Y=y) = g(x)h(y).$"

\begin{enumerate}
\item ``Express $P(X=x)$ in terms of $g$ and $h$."
\item ``Express $P(Y=y)$ in terms of $g$ and $h$."
\item ``Show that $(\sum_x g(x))(\sum_y h(y))=1$."
\item ``Show that $X$ and $Y$ are independent."
[Recall that $X$ and $Y$ are independent if the joint density of $X$ and $Y$ is equal to the product of the densities of $X$ and $Y$:  $f(x,y)=f(x)f(y)$.]
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(Source:  \emph{Introduction to Probability Theory} by Hoel, Port, and Stone, 1971, p. 79.)

\item  Recall the following definition:

\begin{definition}
The random variables $X_1, X_2, X_3, \ldots , X_n$ are \emph{mutually independent} if $$f(x_1, x_2, x_3, \ldots, x_n) = f_1(x_1)f_2(x_2)f_3(x_3) \cdots f_n(x_n),$$ where $f$ is the joint density of $X_1$, $X_2$, $X_3, \ldots$, $X_n$ and $f_i$ is the density of $X_i$.  In other words, random variables are mutually independent if their joint density is the product of the marginal densities.
\end{definition}

Prove that if random variables $X_1, X_2, \ldots, X_n$ are mutually independent, then they are pairwise independent;  that is, $X_i$ and $X_j$ are independent for each pair $i \neq j$.  (Recall that this is not true for events in a sample space in general;  however, it is true for random variables.  Random variables are special because they are functions:  distinct values of a random variable are the images of mutually exclusive events.)

Suggestion:  By renumbering the random variables, we may assume that $i=1$ and $j=2$ (since multiplication is commutative);  this makes the notation much easier.  

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