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\begin{document} 
\title{Assignment 5}
\section{Assignment 5}  
\subsection{Problem 1}
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Suppose $Y$ is a discrete random variable with $E(Y)=\mu$ and $a$ and $b$ are constants.  Use the definition of expected value to show that:
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a) $E(a\cdot Y)=a\cdot E(Y)$
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b) $E(Y+b)=E(Y)+b$
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c) $E(a\cdot Y+b)=a\cdot E(Y)+b$
\subsection{Problem 2}
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Suppose $Y$ is a discrete random variable with $V(Y)=\sigma^2$ and $a$ and $b$ are constants.  Use the definition of variance to show that:
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a) $V(a\cdot Y)=a^2\cdot V(Y)$
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b) $V(Y+b)=V(Y)$
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c) $V(a\cdot Y+b)=a^2\cdot V(Y)$

\subsection{Problem 3}
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Three chips are randomly placed into 3 bins. 
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a) Describe the probability triple $(S,\Sigma,\mu)$ for this experiment
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b) Define a random variable $Y:S\rightarrow\mathbb{R}$ such that for $s\in S$, $Y(s)$ is the number of empty bowls. What is $Y[S]$, the range of $Y$?
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c) Assuming all elements of $S$ are equally likely, define a probability function for $Y$
\[
p(y)=P(Y=y)=\mu\left(Y^{-1}[y]\right),\quad y\in Y[S]
\]
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d) Find $E(Y)$ and $V(Y)$
\subsection{Problem 4}
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Five balls numbered $1$ through $5$ are placed in an urn and two are selected. 
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a) Describe the probability triple $(S,\Sigma,\mu)$ for this experiment
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b) Define a random variable $Y:S\rightarrow\mathbb{R}$ such that for $s\in S$, $Y(s)$ is the \textit{larger} of the two numbers drawn. What is $Y[S]$, the range of $Y$?
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c) Assuming all elements of $S$ are equally likely, define a probability function for $Y$
\[
p(y)=P(Y=y)=\mu\left(Y^{-1}[y]\right),\quad y\in Y[S]
\]
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d) Find $E(Y)$ and $V(Y)$

\end{document}



