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\title{\vspace{-2cm} \sc Math 20 -- Problem Set 5 (due August 1) \vspace{-2.1cm}}
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%{\textbf{Name} \underline{\hspace{10cm}} } \\

This problem set is due at the \emph{beginning} of class.  This is just the problem list; please work out these problems on a different sheet of paper.  Please write neatly, staple the pages together, and explain your work where appropriate.  You do not need to simplify binomial coefficients $\binom{n}{k}$ for both which $k>3$ and $n-k>3$, or exponentials $n^k$ where $n+k >8$. \\

\begin{enumerate}

\item Let $X$ be a random variable with range $[-1, 1]$ and let $f(x)$ be its density of $X$.  Find $\mu_X$ and $\sigma^2_X$, if, for $|x| \leq 1$:

\begin{enumerate}

\item $f(x) = \frac{3}{4}(1-x^2)$

\item $f(x) = \frac{x+1}{2}$

\item $f(x) = \frac{3}{8}(x+1)^2$

\end{enumerate}
(This is \#2 a,c,d from Grinstead and Snell, page 277.)  \\

\item Let $X$ be a random variable defined on the interval $[0, \pi]$ whose density function is $f(x) = \frac{\sin(x)}{2}$.  Compute the CDF $F(x)$.  What is $P(\frac{\pi}{6} \leq X \leq \frac{\pi}{2})$? \\

\item On an average 8-hour school day, 960 people walk into Kemeny Hall.  Assume, though this is certainly not the case, that this happens randomly at a constant rate over the 8 hours.  What is the probability that exactly 8 people walk into Kemeny Hall within a 10-minute interval during the school day?  What is the probability that exactly 48 people walk into Kemeny Hall within an hour? \\

\item The \emph{half-life} of an isotope is the amount of time it takes for the probability of one isotope to decay into another to be 50\%.  The time it takes for a Carbon-14 isotope to decay into a Nitrogen-14 isotope is given by an exponential distribution with expected value estimated at 8267 years.  Find its half-life to the nearest year.  Use a calculator to simplify exponentials and logarithms.  (Note: Decay of radioactive particles is probably the most approprate process to model with the exponential distribution.) \\

\item Suppose the height of an adult male is given by a normal distribution with expected value 70 inches and standard deviation 4 inches.  

\begin{enumerate}

\item Shaquille O'Neal is 83 inches tall.  What proportion of adult males are taller than Shaq?  Use a standard normal distribution table.

\item Darren Sproles is 66 inches tall.  What proportion of adult males are shorter than Darren?  Do you need a normal distribution table to answer this question?

\end{enumerate}

\item Prove that if $X$ is a continuous random variable with range $[x_1, x_2]$ and finite expected value $\mu$, then,
\[ \Var(X) = E(X^2) - \mu^2.\]

(Note: This is true for \emph{all} discrete and continuous RVs with finite mean and variance, and the proof is almost exactly the same!)


\end{enumerate}


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