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Limiting Extreme-Value Distribution

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I need some help with this statistics problem as well as some help understanding limiting distribution and the limiting extreme-value distribution:
Consider a random sample of size n from a distribution with CDF (cumulative distribution function) F(x)=1-1/x if , and zero otherwise.
a) Derive the CDF of the smallest order statistic, X(1n)
b) Find the limiting distribution of X(1n)
c) Find the limiting distribution of Xn(1n)

Consider the CDF above. Find the limiting extreme-value distribution of and compare this result to the results of exercise 7.

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Solution Summary

This solution has step-by-step calculations and clear explanations to derive the CDF of the smallest order statistic and also the limiting distributions. All formulas used and full workings are included.

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7.1 Consider a random sample of size n from a distribution with CDF (cumulative distribution function) F(x)=1-1/x if , and zero otherwise.
a) Derive the CDF of the smallest order statistic,
Solution. Since CDF (cumulative distribution function) F(x)=1-1/x if and zero otherwise, we get the pdf(probability density function) as follows.

Now, we consider the CDF of , denoted by , ...

Solution provided by:
Changping Wang, MA
Education
  • BSc , Wuhan Univ. China
  • MA, Shandong Univ.
Recent Feedback
  • "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
  • "excellent work"
  • "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
  • "Thank you"
  • "Thank you very much for your valuable time and assistance!"
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