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Problem set policies. Please provide concise, clear answers for each question. Note that only writing the result of a calculation (e.g., "SD = 3.3") without explanation is not sufficient. For problems involving R, be sure to include the code in your solution.

Please submit your problem set via Canvas as a PDF, along with the R Markdown source file.

We encourage you to discuss problems with other students (and, of course, with the course head and the TFs), but you must write your final answer in your own words. Solutions prepared "in committee" are not acceptable. If you do collaborate with classmates on a problem, please list your collaborators on your solution.

Discussion Prompt.

For this week, choose between one of the following options:

– Find an example of a named distribution not discussed in the class. Briefly summarize the features of the distribution and how it is used; if you have found your favorite distribution1, feel free to share why you consider it your favorite. Some interesting distributions are the Weibull distribution and Gamma distribution. The second page of this article provides the names of many probability distributions.

– Share an example of how random variables are used in a real-world context. For example, you might enjoy reading about modern portfolio theory2 if you are interested in finance. This article discusses the distribution of the number of births in a single day in England and Wales, this article discusses the methodology behind an election forecasting model, and this article discusses how differential privacy works. For something a bit more light-hearted, this article discusses examining the distribution of times popcorn kernels pop. Briefly summa- rize how random variables are used in your chosen example and comment on what you find interesting.

– Benford’s Law refers to the observation that the leading digit in sets of numerical data fol- lows a particular distribution (that is non-uniform). A number of reports have applied Ben- ford’s Law to COVID-19 data, including this article assessing whether there may be misre- porting of COVID-19 deaths in the US, this article examining COVID-19 statistics in the US and worldwide, and this article looking at the association between a country’s COVID-19 re- porting accuracy and development. Benford’s Law has also been used to detect fraud. Share an idea related to the application of Benford’s Law that you find interesting; feel free to use these linked articles or investigate how Benford’s Law has been used in other contexts.

To receive full participation credit, 1) write a post in the #discussion channel on Slack and 2) respond to someone else’s post. The deadline for posting is the same as the problem set deadline.

1A must for any statistician!

2For a more technical discussion, see this resource from the University of Washington.

Problem 1.

According to data from the CDC, about 37.1% of adults (individuals 18 years of age or older) in the United States and 57.9% of children (individuals between 6 months and 17 years of age) in the United States received a flu vaccine during the 2017-2018 flu season.

For any calculations, be sure to demonstrate the reasoning behind your calculations, such as by defining the relevant random variable(s), using probability notation, and/or briefly writing out your thought process.

a) Consider a random sample of 50 adults from the Boston area.

i. Calculate the probability that exactly 20 adults received a flu vaccine.

ii. Calculate the probability that exactly 30 adults did not receive a flu vaccine.

b) Consider a random sample of 20 children from the Boston area.

i. What is the probability that at most 10 children received a flu vaccine?

ii. What is the probability that at least 11 children received a flu vaccine?

c) State two assumptions you needed to make in order to answer parts a) and b). Briefly com- ment on the extent to which those assumptions were reasonable.

d) Consider a random sample of n = 70 individuals, which consists of 50 adults and 20 chil- dren. Let Z represent the total number of individuals who received the flu vaccine in the sample of 70 individuals. Does Z follow a binomial distribution? Explain your answer.

Problem 2.

Assume the annual returns on a stock portfolio are normally distributed with a mean of 14.7% and a standard deviation of 33%. A return of 0% indicates the value of the portfolio does not change.

For any calculations, be sure to demonstrate the reasoning behind your calculations, such as by defining the relevant random variable(s), using probability notation, and/or briefly writing out your thought process.

a) What is the probability that in any given year the portfolio will lose money?

b) What is the probability that in any given year the portfolio will have at least a 50% return?

c) What is the probability that in any given year the portfolio will have a return between 25% and 75%?

d) Calculate the return value that marks off the lowest 10% of annual returns for this portfolio.

e) What is the probability that four of the next ten years will have a return greater than 50%? Comment on the validity of any assumptions required to make this calculation.

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