DLMDSAS01 Advanced Statistics

ADVANCED WORKBOOK

Task for: DLMDSAS01 Advanced Statistics

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The workbook bases on the output of the parameter generator. The produced numbers are ????₁ to ????₂₀ and are used in the assignment. This gives each student a personal set of numbers. In the tasks below, several parts start with sentences such as If ????₉ is 5: Do not perform tasks if the corresponding value is different in your parameters (e.g., if you have ????₉ = 3).

Task 1: Basic Probabilities and Visualizations (1)

Please provide the requested visualization as well as the numerical results. In both cases, please either prove or cite any computation of the proof steps (calculations, code, steps, etc.), and justify why you trust the tools you used. Do not forget to include the scale of each graphics so that the reader can interpret the numbers represented. As indicated in the remark above, only perform one of the following tasks based on your personal value of ????₁:

If ????₁ is 0: A vote with the outcome ???????????? or ???????????????????????????? follows a Bernoulli distribution where ????(vote = "????????????") = ????₂. Represent the proportion of for and against in this single Bernoulli trial using a graphic and a percentage. Can an expectation be calculated? Justify your answer with all necessary hypotheses.

If ????₁ is between 1 and 3:

The number of meteorites falling into an ocean in a given year can be modeled by one of the following distributions. Provide a graphic showing the probability of one, two, three, etc., meteorites falling (until the probability remains less than 0.5% for any higher number of meteorites; you should also prove it). Calculate the expectation and median and present them in this graphic:

• If ????₁ is 1: a Poisson distribution with an expectation of ???? = ????₂


• If ????₁ is 2: a negative binomial distribution with an expectation of ???? = ????₂ and ???? = ????₃


• If ????₁ is 3: a geometric distribution counting the number of Bernoulli trials with ???? = ????₂ until it

Task 2: Basic Probabilities and Visualizations (2)

Let ???? be the random variable with the time it takes to hear an owl from your rooms open window (in hours). Assume that the probability that you still need to wait to hear the owl after ???? hours is one of the following functions:

• If ????4is 0: the probability is given by ????5e^?????6????+ ????7e^?????8????


• If ???? is 1: the probability is given by ???? e^?????6 ????2 + ???? e^?????8 ????8

4 5 7

• If ???? is 2: the probability is given by ???? e^?????6 ????? + ???? e^????? 3 ????

4 5 7 8 ?

• If ???? is 3: the probability is given by ???? e^?????6 ????2 + ???? e^?????8 ????2

4 5 7

Find the probability that you need to wait between two and four hours to hear the owl, and compute and display the probability density function graph, as well as a histogram where each bar represents the probability of the hearing the owl at any particular minute. Compute and display in the graphics the mean, variance, and quartiles of the waiting times.

Please pay attention to the various units of time and make sure that the parameters you receive are such that

????5 + ????7 = 1.

Task 3: Transformed Random Variables

A type of network router has a bandwidth total to first hardware failure called ???? expressed in terabytes. The random variable ???? is modeled by an exponential distribution whose density is given by one of the following functions:

• If ????₉ = 0: ????????(????)=

1 ????

?

1

?????

????

4

?????

• If ????₉ = 1: ????????(????)=

24? 1

5 ???? ????

6

????

?????

• If ????₉ = 2: ????????(????)= ???? ???? ????

120????7

with a single parameter ????. Consider the bandwidth total to failure ???? of the sequence of the two routers of the same type (one being brought up automatically when the first is broken).

Express ???? in terms of the bandwidth total to failure of single routers ????₁ and ????₂. Formulate realistic assumptions about these random variables. Calculate the density function of the variable ????.

Given an experiment with the dual-router-system yielding a sample ????₁ , ????₂ , , ????_???? , calculate the likelihood function for ????. Propose a transformation of this likelihood function whose maximum is the same and can be computed easily.

An actual experiment is performed, and the infrastructure team has obtained the bandwidth totals to failure given by the sequence ????₁₀ of numbers. Estimate the model-parameter with the maximum likelihood and compute the expectation of the bandwidth total to failure of the dual-router-system.

Task 4: Hypothesis Test

Over a long period of time, the production of 1,000 high-quality hammers in a factory seems to have reached a weight with an average of ????₁₁ (in ????) and standard deviation of ????₁₂ (in ????). Propose a model for the weight of the hammers including a probability distribution for the weight. Provide all the assumptions needed for this model to hold (even the uncertain ones). What parameters does this model have?

One aims to answer one of the following questions about a new production system:

• (if ????₁₃ = 0): Does the new system make more constant weights?


• (if ????₁₃ = 1): Does the new system make lower weights?


• (if ????₁₃ = 2): Does the new system make higher weights?


• (if ????₁₃ = 3): Does the new system make less constant weights?

To answer this question, a random sample of newly produced hammers is evaluated yielding the weights in ????₁₄.

What hypotheses can you propose to test the question? What test and decision rule can you make to estimate whether the new system answers the given question? Express the decision rules as logical statements involving critical values. What error probabilities can you suggest and calculate? Perform the test and draw the conclusion to answer the question

Task 5: Regularized Regression

Given the values of an unknown function ????: ? ? ? at some selected points, we try to calculate the parameters of a model function using OLS as a distance and a ridge regularization:

• (if ????15 = 0): a polynomial model function of twelve ???????? parameters: ????(????) = ????0 + ????1???? + ????2????² + ? +

????12????12

• (if ????15 = 2): a polynomial model function of ten ???????? parameters: ????(????) = ????0 + ????1???? + ????2????² + ? +

????10????10

Calculate the OLS estimate, and the OLS ridge-regularized estimates for the parameters given the sample points of the graph of ???? given that the values are (????, ????) each of the elements of ????₁₆. What weight do you give to the penalties? What are the qualities of each of the solutions?

Remember to include the steps of your computation, which are more important than the actual computations. If you calculate the solution with a program, make sure that you trust and cite the core functions used and that you sketch the mathematical path in a way that is coherent with the program.

Task 6: Bayesian Estimates

Following Hogg et al. (2020), exercise 11.2.2:

Let ????₁, ????₂ , ????₁₀ be a random sample from a gamma distribution with ???? = 3 and ???? = 1/????. Suppose we believe that ???? follows a gamma-distribution with ???? = ????17 and ???? = ????18 and suppose we have a trial (????1, , ????????) with an observed ????? = ????19.

1. Find the posterior distribution of ????.


2. What is the Bayes point estimate of ???? associated with the square-error loss function?


3. What is the Bayes point estimate of ???? using the mode of the posterior distribution?