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# The Pareto distribution is a long-tail probability distribution used to model many social and technical phenomena

INSTRUCTIONS TO CANDIDATES
###### ANSWER ALL QUESTIONS

Q1. Pareto Distribution [8 points]

The Pareto distribution is a long-tail probability distribution used to model many social and technical phenomena (e.g., distribution of wealth, internet traffic). Its probability density function (pdf) is given by:

where xm is the scale parameter and α is the shape parameter.

1. Plot the pdfs of Pareto distributions with α = 2, α = 1 and α = 0.5, while fixing xm = 4. Show the three pdfs in the same figure.

2. Given a set of data {x1, x2, . . . , xn} sampled from a Pareto distribution with known scale parameter xm  = 4,  use  maximum  likelihood  estimation  (MLE)  to  derive  an  estimate  of  the  shape  parameter  αˆ. Clearly show every step of your derivation.

3. Based  on  your  derivation  above,  compute  an  estimated  αˆ  using  the  data  set  in  https://bit.ly/

30U7QVm. The data set is generated from a Pareto distribution model with xm = 4.

4. Using bootstrap resampling (with 1000 resamples), compute the standard error (SE) and 95% confi- dence interval (CI) of the parameter α.

Q2. Weibull Distribution [8 points]

The Weibull distribution is a heavy-tail probability distribution often used to model variable failure rates when analyzing the reliability of complex systems. Its probability density function (pdf) is given by:

f (x) = k   x  k−1 e−( x )k    for x ≥ 0

λ λ

where k is the shape parameter and λ is the scale parameter.

1. Plot the pdfs of Weibull distributions with k = 2, k = 1 and k = 0.5, while fixing λ = 4. Show the three pdfs in the same figure.

2. Given a set of data {x1, x2, . . . , xn} sampled from a Weibull distribution with known shape parameter k = 2, use maximum likelihood estimation (MLE) to derive an estimate of the scale parameter λ. Clearly show every step of your derivation.

3. Based  on  your  derivation  above,  compute  an  estimated  λˆ

using the data set in https://bit.ly/

3HdEaCj. Do the same for another data set in https://bit.ly/30WcS3R. Both data sets are generated from Weibull models with k = 2.

4. Use permutation resampling (with 1000 resamples) to determine if the two models (that generated the two data sets above) share the same parameter λ.  Use one-tailed p-value with a significance level of 5%.

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