Mar 12, 2024

1: Basic Probabilities and Visualizations

Please provide the requested visualization as well as the numeric results. In both cases, please provide how you realized these (calculations, code, steps…) and why it is the appropriate tools. Do not forget to include the scale of each graphics so a reader can read the numbers represented.

• If ξ1 is 0: A vote with outcome for against follows a Bernoulli distribution where p (vote = For ) = ξ 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 by all necessary hypotheses.
• If ξ1 is between 1 and 3: The number of meteorites falling on an ocean in a given year can be modelled by one of the following distributions. Give a graphic showing the probability of one, two, or three… meteorites falling (until the probability remains provably less than 0.5 % for any bigger number of meteorites). Calculate the expectation and median and show them graphically on this graphic:
• If ξ1 is 1: a Poisson distribution with an expectation of Ú = ξ2
• If ξ1 is 2: a negative binomial distribution with number of successes of K = ξ2 und p = ξ3.
• If ξ1 is 3: a geometric distribution counting the number of Bernoulli trials with p = ξ2 until it succeeds.

2. Basic Probabilities and Visualizations (2)

Let y be the random variable with the time to hear an owl from your room’s open window (in hours). Assume that the probability that you still need to wait to hear the owl after y hours is one of the following:

• If ξ4 is 0: the probability is given byξ5e −ξ6 y + ξ7e −ξ8 y
• If ξ4 is 1: the probability is given by ξ5e −ξ6 y2 + ξ7e −ξ8 y 8
• If ξ4 is 2: the probability is given by ξ5e −ξ6 y + ξ7e −ξ8 y3
• If ξ4 is 3: the probability is given by ξ5e −ξ6 y 2 +ξ7e −ξ8 y2

Find the probability that you need to wait between 2 and 4 hours to hear the owl, compute and display the probability density function graph as well as a histogram by the 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.

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 modelled by a distribution whose density is given by one of the following functions:

• (if ξ9 = 0): fs (s) = 1Ø e −s Ø
• (if ξ9 = 1): fs (s) = 1 24Ø 5 s 4 e − s Ø
• (if ξ9 = 2): fs (s) = 1 Ø for s ∈ [0, Ø]

with a single parameter Ø. Consider the bandwidth total to failure T of the sequence of the two routers of the same type (one being brought up automatically when the first is broken). Express T in terms of the bandwidth total to failure of single routers and. 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 T1 , T2, Tn, calculate the likelihood function for Propose a transformation of this likelihood function whose maximum is the same and can be computed easily.

4: Hypothesis Test

Over a long period of time, the production of 1000 high-quality hammers in a factory seems to have reached a weight with an average of ξ11 (in ) and standard deviation of ξ12 (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 at answering one of the following questions about a new production system:

• (if ξ13 = 0): Does the new system make more constant weights?
• (if ξ13 = 1): Does the new system make lower weights?
• (if ξ13 = 2): Does the new system make higher weights?
• (ifξ13 = 3): Does the new system make less constant weights?

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

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

• (if ξ15 = 0): a polynomial model function of thirteen ai parameters: f(x) = Α0 α1x α2x 2 ⋯ α12x 12
• (ifξ15 = 2): a polynomial model function of eleven ai  parameters: f(x) = Α0 α1x α2x 2 ⋯ α12x 12 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 y = ????16. Provide a graphical representation of the graphs of the approximating functions and the data points. Remember to include the steps of your computation which are more important than the actual computations.

6: Bayesian Estimates

(following Hogg, McKean & Craig, exercise 11.2.2) Let x1, x2, …, x10 be a random sample from a gamma distribution with a = 3 and b = 1/Ø. Suppose we believe that Ø follows a gamma-distribution with a = ξ17 and b = ξ18 and suppose we have a trial ( x1 , … , xn ) with an observed x = ξ19.

a) Find the posterior distribution of Ø.

b) What is the Bayes point estimate of Ø associated with the square-error loss function?

c) What is the Bayes point estimate of Ø using the mode of the posterior distribution?

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