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\begin{document} 
\title{Assignment 6}
\section{Assignment 6}  
\subsection{Problem 1}
\par\vspace{0.3 cm}
If $X$ and $Y$ are random variables, the \textbf{covariance} of $X$ and $Y$, denoted by $\sigma_{XY}$ or $\mbox{Cov}(X,Y)$ is defined as:
\[
\sigma_{XY}=\mbox{Cov}(X,Y)=E(XY)-E(X)\cdot E(Y)
\]
where $E(XY)$ is defined as:
\[
E(XY)=\sum_x\sum_y xy\cdot P(X=x\mbox{ and }Y=y)
\]
In a binomial experiment with four trials and probability of success $p$, let $Y$ be the number of successes and $X$ be the number of failures.  
\par\vspace{0.3 cm}
a) What are the values of $E(X)$ and $E(Y)$?
\par\vspace{0.3 cm}
b) What is $E(XY)$?
\par\vspace{0.3 cm}
c) What is $\mbox{Cov}(X,Y)$?
\subsection{Problem 2}
\par\vspace{0.3 cm}
Suppose $(S,\Sigma,\mu)$ is a probability triple associated with an experiment and $Y$ is a random variable defined on that triple.  Show that if we conduct two replications of the experiment in such a way that 
\[
P(Y_1=y_1\mbox{ and }Y_2=y_2)=P(Y=y_1)\cdot P(Y=y_2)
\]
where $Y_1$ and $Y_2$ are the values of the random variable $Y$ associated with the first and second replications, then
\[
\mbox{Cov}(Y_1,Y_2)=0
\]

\subsection{Problem 3}
\par\vspace{0.3 cm}
Recall that the cumulative distribution function of a random variable $Y$ is defined as:
\[
F(y)=P(Y\leq y)
\]
Suppose a random variable $Y$ associated with an experiment has probability function $p(y)$ and cumulative distribution function $F(y)$.  If five independent replications of the experiment are performed, find the cumulative distribution function of the \textit{median} of the five $Y$ values, that is, the third value in the list if the $Y$ values are ordered from largest to smallest.  (hint: pick a value $y_0$ and divide the range of $Y$ into "successes" where $Y_i>y_0$ and "failures" where $Y_i\leq y_0$, and try to define the event "the median is less than or equal to $y_0$ in terms of successes and failures).
\subsection{Problem 4}
\par\vspace{0.3 cm}
Let $(S,\Sigma,\mu)$ be a probability triple and $A,B\in\Sigma$.  Define a random variable
\[
I_A:S\rightarrow\mathbb{R}
\]
by:
\[
I_A(s)=\left\{\begin{array}{lcl}
1 &\mbox{if}& s\in A\\
0 &\mbox{if}& s\notin A\\
\end{array}\right.
\]
and a random vaiable $I_B:S\rightarrow\mathbb{R}$ by: 
\[
I_B:S\rightarrow\mathbb{R}
\]
by:
\[
I_B(s)=\left\{\begin{array}{lcl}
1 &\mbox{if}& s\in B\\
0 &\mbox{if}& s\notin B\\
\end{array}\right.
\]
\par\vspace{0.3 cm}
In terms of elements of $\Sigma$ and their associated pobability measures, 
\par\vspace{0.3 cm}
a) How would you interpret $E(I_A)$?
\par\vspace{0.3 cm}
b) How would you intepret $E(I_A\cdot I_B)$?
\par\vspace{0.3 cm}
c) How would you intepret $E(1-I_A)$?
\par\vspace{0.3 cm}
d) How would you intepret $E[(1-I_A)(1-I_B)]$?
\end{document}



