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\title{\sc Math 20 -- Homework 5}
\author{due Wednesday, August 2}
\date{}
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\textbf{Instructions:} This assignment is due at the \emph{beginning} of class. Staple your work together (do not just fold over the corner). Please write the questions in the correct order. If I cannot read your handwriting, you won't receive full credit.\\
\emph{You may use Wolfram Alpha to compute any necessary sums or integrals.}\\
\textbf{If you're using facts about distributions to answer the questions, be very clear about which distribution you're using to model that problem and why that distribution is appropriate.}
\begin{enumerate}
\item When you listen to your ``Math Homework'' playlist on shuffle on Spotify, you usually hear your favorite song about once every two days. If you then go a whole week without hearing it, how surprised are you? (In other words, what's the probability of this occurring?)
\item On an average 8-hour school day, 1000 people walk into Kemeny Hall. Assume this happens completely randomly\footnote{Of course, this is a terrible assumption---people are more likely to arrive in the short periods between classes. But let's ignore that for now.}. What is the probability that exactly six people enter Kemeny Hall in a ten minute span?
\item Let $X_1, X_2, \ldots, X_k$ be $k$ random variables that are mutually independent and uniformly distributed on the interval $[0,1]$. Define a new random variable $Y = \min(X_1, X_2, \ldots, X_k)$ such that the value of $Y$ is the smallest of the values of $X_1, X_2, \ldots, X_k$. Find $\E[Y]$.
\item Let $X$ be a discrete random variable that takes only positive integer values. Our normal formula for the expected value of $X$ says
\[
\E[X] = \sum_{k=1}^\infty kP(X=k).
\]
Prove the following alternate formula:
\[
\E[X] = \sum_{k=1}^\infty P(X \geq k).
\]
\end{enumerate}
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