Hypothesis: Central limit theorem shows the sample size >= 30 can applies the central limit theorem. Methodology: Probability distribution simulator, simulation Findings: Section 3 shows the random samples of different probability distribution have different sample size to apply the property of central limit theorem. Contribution: Statistics cannot be the tool of the big data analysis and artificial intelligence. New Central limit theorem can be the basis of Statistics applied in Big data and Artificial intelligence.

1. Problems using Central
Limit Theorem applied
Mei-Yu Lee (Writer),
Keng-Wei Wang (Data collector),
Kuan-Sian Wang (Program designer)

2. Methodology from
• The E-book has been published on
Amazon.
• See the link at the video description or
scan the QR code.

3. Contents
1. Introduce central limit theorem (CLT)
2. The minimum sample size when CLT applied
2.1. Normal distribution
2.2. Uniform distribution
2.3. Shifted exponential distribution
2.4. Pareto distribution (including Pareto 1 and Pareto 2)
2.5. Arcsin distribution
2.6. Bernoulli distribution
2.7. Rayleigh distribution
2.8. Gumbel distribution
2.9. Logistic distribution
2.10. Weibull distribution
3. CLT requirements for different population distribution

4. 1. Introduce Central Limit Theorem

5. 1. Introduce Central Limit Theorem

6. 1. Introduce Central Limit Theorem

7. 2. The minimum sample size when CLT
applied

8. 2.1. Normal distribution

9. 2.1. Normal distribution

10. 2.2. Uniform distribution

11. 2.2. Uniform distribution

12. 2.3. Shifted exponential distribution

13. 2.3. Shifted exponential distribution

14. 2.4. Pareto 1 distribution

15. 2.4. Pareto 1 distribution

16. 2.4.2. Pareto 2 distribution

17. 2.4.2. Pareto 2 distribution

18. 2.5. Arcsin distribution

19. 2.5. Arcsin distribution

20. 2.6. Bernoulli distribution
Let sample size=30.
The sampling distribution of Z score
of sample mean is difference when p
changed.

21. 2.6. Bernoulli distribution
Let sample size=30.
The sampling distribution of Z score
of sample mean is difference when p
changed.

22. 2.6. Bernoulli distribution
Let sample size=30.
The sampling distribution of Z score
of sample mean is difference when p
changed.

23. 2.6. Bernoulli distribution

24. 2.6. Bernoulli distribution

25. 2.6. Bernoulli distribution

26. 2.6. Bernoulli distribution

27. 2.6. Bernoulli distribution

28. 2.6. Bernoulli distribution

29. 2.7. Rayleigh distribution

30. 2.7. Rayleigh distribution

31. 2.8. Gumbel distribution

32. 2.8. Gumbel distribution

33. 2.9. Logistic distribution

34. 2.9. Logistic distribution

35. 2.10. Weibull distribution

36. 2.10. Weibull distribution

37. 2.10. Weibull distribution
Change
gamma

38. 2.10. Weibull distribution

39. 3. CLT requirements for different
population distribution
Please see “YouTube” about the following population distribution
and which the requirement of CLT applied.
https://www.youtube.com/playlist?list=PLGIquq4uoXLW1n8dU9P_Nh
mXrVvlU5iE2
There are five methods to explain the process, happening reasons,
and minimum sample size of applied CLT.

40. 3.1. List of 28 cases
Case 1, The population distribution ~ Normal distribution,
Case 2, The population distribution~ Uniform distribution,
Case 3, The population distribution ~ Shifted exponential distribution,
Case 4, The population distribution ~ Pareto 1 distribution,
Case 5, The population distribution ~ Arcsin distribution,
Case 6, The population distribution ~ Bernoulli distribution,
(one population proportion)
Case 7, The population distribution ~ Rayleigh distribution,
Case 8, The population distribution ~ Gumbel distribution,
Case 9, The population distribution ~ Logistic distribution,
Case 10, The population distribution ~ Weibull distribution,

41. 3.1. List of 28 cases
Case 11, The population distribution ~ Pareto 2 distribution,
Case 12, The population distribution ~ Pareto 3 distribution,
Case 13, The population distribution ~ Triangular 2 distribution,
Case 14, The population distribution ~ Triangular 3 distribution,
Case 15, The population distribution ~ Log logistic distribution,
Case 16, The population distribution ~ Hyperbolic secant distribution,
Case 17, The population distribution ~ Kumaraswamy distribution,
Case 18, The population distribution ~ Gumbel(type 1) distribution,
Case 19, The population distribution ~ Gumbel(type 2) distribution,
Case 20, The population distribution ~ Double exponential distribution,

42. 3.1. List of 28 cases
Case 21, The population distribution ~ Continuous Bernoulli distribution,
Case 22,The population distribution ~ Generalized logistic distribution type I,
Case 23, The population distribution ~ Exponential logarithmic distribution,
Case 24, The population distribution ~ Dagum distribution,
Case 25, The population distribution ~ Gompertz distribution,
Case 26, The population distribution ~ U quadratic distribution,
Case 27, The population distribution ~ Semicircle distribution,
Case 28, The population distribution ~ Discrete Uniform distribution

43. 3.2. The minimum sample size
requirement when CLT applied
• Case 1, The population distribution ~ Normal distribution,
• the random samples summation is Normal distribution, it is not CLT.
• Case 2, The population distribution~ Uniform distribution,
• the sample size ≥ 20 when CLT applied.
• Case 3, The population distribution ~ Shifted exponential distribution,
• the sample size ≥ 1000 when CLT applied.
• Case 4, The population distribution~ Pareto 1 distribution,
• the sample size ≥ 500 when CLT applied and lambda = 10,
• the minimum sample size requirement is affected by the parameter lambda.
• Case 5, The population distribution~ Arcsin distribution,
• the sample size ≥ 50 when CLT applied.

44. 3.2. The minimum sample size
requirement when CLT applied
• Case 6, The population distribution~ Bernoulli distribution, (one population proportion)
• the sample size ≥ 1000 when CLT applied and p = 0.5,
• the minimum sample size requirement is affected by the parameter p (population proportion).
• Case 7, The population distribution~ Rayleigh distribution,
• the sample size ≥ 100 when CLT applied.
• Case 8, The population distribution~ Gumbel distribution,
• the sample size> ≥ 300 when CLT applied.
• Case 9, The population distribution~ Logistic distribution,
• the sample size ≥ 30 when CLT applied.
• Case 10, The population distribution~ Weibull distribution,
• the sample size ≥ 15 when CLT applied and gamma = 3.5,
• the minimum sample size requirement is affected by the parameter gamma.

45. 3.2. The minimum sample size
requirement when CLT applied
• Case 11, The population distribution ~ Pareto 2 distribution,
• the sample size ≥ 1000 when CLT applied and lambda = 5,
• the minimum sample size requirement is affected by the parameter lambda.
• Case 12, The population distribution ~ Pareto 3 distribution,
• the sample size ≥ 350 when CLT applied and lambda = 5,
• the minimum sample size requirement is affected by the parameter lambda.
• Case 13, The population distribution ~ Triangular 2 distribution,
• the sample size ≥ 80 when CLT applied and a = -10, b = 20, c = -5
• the minimum sample size requirement is affected by the parameters a, b, c.

46. 3.2. The minimum sample size
requirement when CLT applied
• Case 14, The population distribution ~ Triangular 3 distribution,
• the sample size ≥ 150 when CLT applied and a = -10, b = 20, c = -5
• the minimum sample size requirement is affected by the parameters a, b, c.
• Case 15, The population distribution ~ Log logistic distribution,
• the sample size ≥ 1000 when CLT applied and beta = 8,
• the minimum sample size requirement is affected by the parameters beta.
• Case 16, The population distribution ~ Hyperbolic secant distribution,
• the sample size ≥ 40 when CLT applied.
• Case 17, The population distribution ~ Kumaraswamy distribution,
• the sample size ≥ 200 when CLT applied and a = 5, b = 2
• the minimum sample size requirement is affected by the parameters a, b.

47. 3.2. The minimum sample size
requirement when CLT applied
• Case 18, The population distribution ~ Gumbel(type 1) distribution,
• the sample size ≥ 200 when CLT applied and a = 10, b = 8
• the minimum sample size requirement is affected by the parameters a, b.
• Case 19, The population distribution ~ Gumbel(type 2) distribution,
• the sample size ≥ 5000 when CLT applied and a = 5,
• the minimum sample size requirement is affected by the parameters a.
• Case 20, The population distribution ~ Double exponential distribution,
• the sample size ≥ 100 when CLT applied.
• Case 21, The population distribution ~ Continuous Bernoulli distribution,
• the sample size ≥ 60 when CLT applied and lambda = 0.7,
• the minimum sample size requirement is affected by the parameter lambda.

48. 3.2. The minimum sample size
requirement when CLT applied
• Case 22,The population distribution ~ Generalized logistic distribution type I,
• the sample size ≥ 2060 when CLT applied and alpha = 3,
• the minimum sample size requirement is affected by the parameter alpha.
• Case 23, The population distribution ~ Exponential logarithmic distribution,
• the sample size ≥ 10000 when CLT applied and p = 0.4, beta = 4
• the minimum sample size requirement is affected by the parameters p, beta.
• Case 24, The population distribution ~ Dagum distribution,
• the sample size ≥ 30000 when CLT applied and a = 2, b = 1, p = 2.5
• the minimum sample size requirement is affected by the parameters a, b, p.

49. 3.2. The minimum sample size
requirement when CLT applied
• Case 25, The population distribution ~ Gompertz distribution,
• the sample size ≥ 250 when CLT applied and beta = 4,
• the minimum sample size requirement is affected by the parameter beta.
• Case 26, The population distribution ~ U quadratic distribution,
• the sample size ≥ 40 when CLT applied.
• Case 27, The population distribution ~ Semicircle distribution,
• the sample size ≥ 30 when CLT applied.
• Case 28, The population distribution ~ Discrete Uniform distribution
• the sample size ≥ 30 when CLT applied and N = 10.

50. 3.3. Summary
Symmetric distribution Minimal sample size Parameter condition
Triangular 2 80 a = -10, b = 20, c = -5
Continuous Bernoulli 60 λ = 0.7
Arcsin 50
Hyperbolic secant 40
U quadratic 40
Logistic 30
Semicircle 30
Discrete Uniform 30 N=10
Uniform 20
Weibull 15 γ = 3.5

51. 3.3. Summary
Distribution Minimal sample size
Pareto 1 500 λ = 10
Pareto 3 350 λ = 5
Gumbel 300
Gompertz 250 beta = 4
Kumaraswamy 200 a = 5, b = 2
Gumbel(type 1) 200 a = 10, b = 8
Triangular 3 150 a = -10, b = 20, c = -5
Double exponential 100
Rayleigh 100

52. 3.3. Summary
Distribution Minimal sample size
Dagum 30000 a = 2, b = 1, p = 2.5
Exponential logarithmic 10000 p = 0.4, beta = 4
Gumbel(type 2) 5000 a = 5
Generalized logistic type I 2060 alpha = 3
Log logistic 1000 beta = 8
Pareto 2 1000 λ = 5
Bernoulli 1000 p = 0.5
Shifted exponential 1000

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