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\noindent {\large  \textbf{Stat 110 Homework 8, Fall 2017}} 

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\noindent \textbf{Due}: Friday 11/10 at 5:00 pm, submitted as a PDF via the  \href{https://canvas.harvard.edu/courses/27764}{{course webpage}}. Please check carefully to make sure you upload the correct file. Your submission must be a single PDF file, no more than $20$ MB in size. It can be typeset or scanned, but must be clear and easily legible (not blurry or faint) and correctly rotated. No submissions on paper or by email will be accepted. Please show your work and give clear, careful, convincing justifications. See the syllabus for the collaboration policy. 

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\noindent 1. (BH 8.1) Find the PDF of $e^{-X}$ for $X \sim \Expo(1)$.

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\noindent 2. (BH 8.12) Three students are working independently on their probability homework. They start at the same time. The times that they take to finish it are i.i.d.~random variables $T_1,T_2,T_3$ with $T_j^{1/\beta} \sim \Expo(\lambda)$, where $\beta$ and $\lambda$ are known positive constants.

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\noin (a) Find the PDF of $T_1$. 

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\noin (b) Find expressions for $E(T_1^2)$ as integrals in two different ways, one based on the PDF of $T_1$, and the other based on the $\Expo(\lambda)$ PDF (do not simplify).

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\noindent 3. (BH 8.13)  Let $T$ be the ratio $X/Y$ of two i.i.d.~$\N(0,1)$ r.v.s.~$X,Y$. This is the \emph{Cauchy} distribution and, as shown in Example 7.1.24, it has PDF
$$f_T(t) = \frac{1}{\pi (1+t^2)}.$$

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\noin (a) Show that $1/T$ has the same distribution as $T$ using calculus, after first finding the CDF of $1/T$. (Note that the one-dimensional change of variables formula does not apply directly, since the function $g(t)=1/t$, even though it has $g'(t)<0$ for all $t \neq 0$, is undefined at $t=0$ and is not a strictly decreasing function on its domain.)

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\noin (b) Show that $1/T$ has the same distribution as $T$ without using calculus, in 140 characters or fewer. 

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\noindent 4. (BH 8.37)  Let $X \sim \Pois(\lambda t)$ and $Y \sim \Gam(j, \lambda)$, where $j$ is a positive integer. Show using a story about a Poisson process that \[P(X \ge j) = P(Y \le t).\]

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\noindent 5. (BH 8.38)  Visitors arrive at a certain scenic park according to a Poisson process with rate $\lambda$ visitors per hour. Fred has just arrived (independent of anyone else), and will stay for an $\Expo(\lambda_2)$ number of hours.  Find the distribution of the number of other visitors who arrive at the park while Fred is there.

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\noin Hint: Consider and apply Story 8.4.5 in the book.

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\noindent 6. (BH 8.39) 

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\noindent (a) Let $p \sim \Beta(a,b)$, where $a$ and $b$ are positive real numbers. Find $E(p^2(1-p)^2)$, fully simplified ($\Gamma$ should not appear in your final answer).

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Two teams, $A$ and $B$, have an upcoming match. They will play five games and the winner will be declared to be the team that wins the majority of games. Given $p$, the outcomes of games are independent, with probability $p$ of team $A$ winning and $1-p$ of team $B$ winning. But you don't know $p$, so you decide to model it as an r.v., with $p \sim \Unif(0,1)$ a priori (before you have observed any data).

To learn more about $p$, you look through the historical records of previous games between these two teams, and find that the previous outcomes were, in chronological order, $AAABBAABAB$. (Assume that the true value of $p$ has not been changing over time and will be the same for the match, though your \emph{beliefs} about $p$ may change over time.)

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\noin (b) Does your posterior distribution for $p$, given the historical record of games between $A$ and $B$, depend on the specific order of outcomes or only on the fact that $A$ won exactly $6$ of the $10$ games on record? Explain.

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\noin (c) Find the posterior distribution for $p$, given the historical data.

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The posterior distribution for $p$ from (c) becomes your new prior distribution, and the match is about to begin!

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\noin (d) Conditional on $p$, is the indicator of $A$ winning the first game of the match positively correlated with, uncorrelated with, or negatively correlated of the indicator of $A$ winning the second game of the match? What about if we only condition on the historical data?

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\noin (e) Given the historical data, what is the expected value for the probability that the match is not yet decided when going into the fifth game (viewing this probability as an r.v.~rather than a number, to reflect our uncertainty about it)?

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\noindent 7. (BH 8.47) Let $X_1,X_2,\dots$ be i.i.d.~r.v.s with CDF $F$, and let $M_n = \max(X_1,X_2,\dots,X_n)$. Find the joint distribution of $M_n$ and $M_{n+1}$, for each $n \geq 1$.

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\noin Hint: Consider the cases $a \leq b$ and $a > b$ separately in $P(M_n \leq a, M_{n+1} \leq b)$.


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