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\theoremstyle{plain}    
\newtheorem*{definition}{Definition}
\newtheorem*{example}{Example}
\newtheorem*{exercise}{Exercise}
\newtheorem*{conjecture}{Conjecture}
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\newtheorem*{lemma}{Lemma}
\newtheorem*{remark}{Remark}
\begin{document} 
\title{Written Assignment 1}
\section{Written Assignment 1}
The coin toss experiment has two outcomes, H and T. In the following problems, the experiment consists of repeating the coin toss three times.  
\par\vspace{0.3 cm}
The assignment is to construct a probability space $S=(\Omega,\mathcal{F},\rho)$ that represents the experiment, and to define a random variable $X$ on it.  
\subsection{Problem 1}
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What is the sample space $\Omega$ for this experiment?
\subsection{Problem 2}
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Define a set of events $\mathcal{F}$.  You can define this in many ways, but the logical choice would be something similar to what we did in class.  At the very least $\mathcal{F}$ should include $\Omega$ itself and each outcome contained in $\Omega$.
\subsection{Problem 3}
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For your choice of $\mathcal{F}$, define a probability measure, that is, a function 
\[
\rho: \mathcal{F}\rightarrow[0,1]
\vspace{0.3 cm}
\]
that is consistent with the Kolmogorov axioms.
\subsection{Problem 4}
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Define a random variable on $\Omega$, that is, a function
\[
X: \Omega\rightarrow\mathbb{R}
\]
There are many ways to do this.  You should express $X$ as a table of ordered pairs, the first being an element of $\Omega$ and the second a real number of your choosing.
\subsection{Problem 5}
State the value of the probability that $X$ assumes each value in its range in this probability space $S=(\Omega,\mathcal{F},\rho)$, that is,
\[
P(X=x)\quad\mbox{for each}\quad x\in X[\Omega]
\vspace{0.5 cm}
\]
where $X[\Omega]$ represents the image of $\Omega$ under the function $X$ (which is just a precise way of saying all of the 
values that your random variable can assume). 

\end{document}



