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\noindent {\large  \textbf{Stat 110 Homework 4, Fall 2017}} 

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\noindent \textbf{Due}: Friday 10/6 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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\noin 1. (BH 4.10) Consider the St.~Petersburg paradox (Example 4.3.13), except that you receive $\$n$ rather than $\$2^n$ if the game lasts for $n$ rounds. What is the fair value of this game? What if the payoff is $\$n^2$?

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\noin 2. (BH 4.14)  Let $X$ have PMF 
$$P(X=k) = c p^k/k \textrm{ for $k=1,2,\dots$},$$ where $p$ is a parameter with $0<p<1$ and $c$ is a normalizing constant. We have $c=-1/\log(1-p)$, as seen from the Taylor series 
$$-\log(1-p) = p + \frac{p^2}{2} + \frac{p^3}{3} + \dots.$$ This distribution is called the \emph{Logarithmic} distribution (because of the log in the above Taylor series), and has often been used in ecology. Find the mean and variance of $X$. 

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\noin 3. (BH 4.16) The dean of Blotchville University boasts that the average class size there is $20$. But the reality experienced by the majority of students there is quite different: they find themselves in huge courses, held in huge lecture halls, with hardly enough seats or Haribo gummi bears for everyone. The purpose of this problem is to shed light on the situation. For simplicity, suppose that every student at Blotchville University takes only one course per semester.

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\noin (a) Suppose that there are $16$ seminar courses, which have $10$ students each, and $2$ large lecture courses, which have $100$ students each. Find the dean's-eye-view average class size (the simple average of the class sizes) and the student's-eye-view average class size (the average class size experienced by students, as it would be reflected by surveying students and asking them how big their classes are). Explain the discrepancy intuitively.

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\noin (b) Give a short proof that for \emph{any} set of class sizes (not just those given above), the dean's-eye-view average class size will be strictly less than the student's-eye-view average class size, unless all classes have exactly the same size. 

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\noin Hint: Relate this to the fact that variances are nonnegative.


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\noin 4. (BH 4.25) Nick and Penny are independently performing independent Bernoulli trials. For concreteness, assume that Nick is flipping a nickel with probability $p_1$ of Heads and Penny is flipping a penny with probability $p_2$ of Heads. Let $X_1,X_2,\dots$ be Nick's results and $Y_1,Y_2,\dots$ be Penny's results, with $X_i \sim \Bern(p_1)$ and $Y_j \sim \Bern(p_2)$. 

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\noin (a) Find the distribution and expected value of the first time at which they are simultaneously successful, i.e., the smallest $n$ such that $X_n=Y_n=1$.

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\noin Hint: Define a new sequence of Bernoulli trials and use the story of the Geometric.

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\noin (b) Find the expected time until at least one has a success (including the success).

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\noin Hint: Define a new sequence of Bernoulli trials and use the story of the Geometric.

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\noin (c) For $p_1=p_2$, find the probability that their first successes are simultaneous, and use this to find the probability that Nick's first success precedes Penny's.


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\noin 5. (BH 4.28)  In many problems about modeling count data, it is found that values of zero in the data are far more common than can be explained well using a Poisson model (we can make $P(X=0)$ large for $X \sim \Pois(\lambda)$ by making $\lambda$ small, but that also constrains the mean and variance of $X$ to be small since both are $\lambda$). The \emph{Zero-Inflated Poisson} distribution is a modification of the Poisson to address this issue, making it easier to handle frequent zero values gracefully.

A Zero-Inflated Poisson r.v.~$X$ with parameters $p$ and $\lambda$ can be generated as follows. First flip a coin with probability of $p$ of Heads. Given that the coin lands Heads, $X=0$. Given that the coin lands Tails, $X$ is distributed $\Pois(\lambda)$.  Note that if $X=0$ occurs, there are two possible explanations: the coin could have landed Heads (in which case the zero is called a \emph{structural zero}), or the coin could have landed Tails but the Poisson r.v.~turned out to be zero anyway.

For example, if $X$ is the number of chicken sandwiches consumed by a random person in a week, then $X=0$ for vegetarians (this is a structural zero), but a chicken-eater could still have $X=0$ occur by chance (since they might not happen to eat any chicken sandwiches that week). 

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\noin (a) Find the PMF of a Zero-Inflated Poisson r.v.~$X$.

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\noin (b) Explain why $X$ has the same distribution as $(1-I)Y$, where $I \sim \Bern(p)$ is independent of $Y \sim \Pois(\lambda)$. 

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\noin (c) Find the mean of $X$ in two different ways: directly using the PMF of $X$, and using the representation from (b). For the latter, you can use the fact (which we prove in Chapter 7) that if r.v.s $Z$ and $W$ are independent, then $E(ZW)=E(Z)E(W)$. 

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\noin (d) Find the variance of $X$.


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\noin 6. (BH 4.37) You have a well-shuffled 52-card deck. On average, how many pairs of adjacent cards are there such that both cards are red? 



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\noin 7. (BH 4.55) Elk dwell in a certain forest. There are $N$ elk, of which a simple random sample of size $n$ is captured and tagged (so all ${N \choose n}$ sets of $n$ elk are equally likely). The captured elk are returned to the population, and then a new sample is drawn. This is an important method that is widely used in ecology, known as \emph{capture-recapture}. If the new sample is also a simple random sample, with some fixed size, then the number of tagged elk in the new sample is Hypergeometric.

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\noin For this problem, assume that instead of having a fixed sample size, elk are sampled one by one without replacement until $m$ tagged elk have been recaptured, where $m$ is specified in advance (of course, assume that $1 \leq m \leq n \leq N$). An advantage of this sampling method is that it can be used to avoid ending up with a very small number of tagged elk (maybe even zero), which would be problematic in many applications of capture-recapture. A disadvantage is not knowing how large the sample will be.

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\noin (a) Find the PMFs of the number of untagged elk in the new sample (call this $X$) and of the total number of elk in the new sample (call this $Y$). 

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\noin Hint: What does the event $X=k$ say about how many tagged and how many untagged elk there are in the first $m+k-1$ elk sampled? What does it say about the $(m+k)$th elk sampled? 

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\noin (b) Find the expected sample size $EY$ using symmetry, linearity, and indicator r.v.s. 

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\noin Hint: We can assume that even after getting $m$ tagged elk, they continue to be captured until all $N$ of them have been obtained; briefly explain why this can be assumed. Express $X=X_1+\dots + X_m$, where $X_1$ is the number of untagged elk before the first tagged elk, $X_2$ is the number between the first and second tagged elk, etc. Then find $EX_j$ by creating the relevant indicator r.v.~for each untagged elk in the population.

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\noin (c) Suppose that $m,n,N$ are such that $EY$ is an integer. If the sampling is done with a fixed sample size equal to $EY$ rather than sampling until exactly $m$ tagged elk are obtained, find the expected number of tagged elk in the sample. Is it less than $m$, equal to $m$, or greater than $m$ (for $n<N$)?


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