I need some help with this statisticsproblem as well as some help understanding limiting distribution and the limiting extreme-value distribution:

Consider a random sampleof 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.

Please see the attached file.

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 , …

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.