Engineering Statistics Handbook Episode 2 Part 4 docx

1.3.6.6.12. Double Exponential Distribution http://www.itl.nist.gov/div898/handbook/eda/section3/eda366c.htm (7 of 7) [5/1/2006 9:58:09 AM] Cumulative Distribution Function The formula for the cumulative distribution function of the power normal distribution is where is the cumulative distribution function of the standard normal distribution. The following is the plot of the power normal cumulative distribution function with the same values of p as the pdf plots above. 1.3.6.6.13. Power Normal Distribution http://www.itl.nist.gov/div898/handbook/eda/section3/eda366d.htm (2 of 7) [5/1/2006 9:58:10 AM] Percent Point Function The formula for the percent point function of the power normal distribution is where is the percent point function of the standard normal distribution. The following is the plot of the power normal percent point function with the same values of p as the pdf plots above. 1.3.6.6.13. Power Normal Distribution http://www.itl.nist.gov/div898/handbook/eda/section3/eda366d.htm (3 of 7) [5/1/2006 9:58:10 AM] Hazard Function The formula for the hazard function of the power normal distribution is The following is the plot of the power normal hazard function with the same values of p as the pdf plots above. 1.3.6.6.13. Power Normal Distribution http://www.itl.nist.gov/div898/handbook/eda/section3/eda366d.htm (4 of 7) [5/1/2006 9:58:10 AM] Cumulative Hazard Function The formula for the cumulative hazard function of the power normal distribution is The following is the plot of the power normal cumulative hazard function with the same values of p as the pdf plots above. Survival Function The formula for the survival function of the power normal distribution is The following is the plot of the power normal survival function with the same values of p as the pdf plots above. 1.3.6.6.13. Power Normal Distribution http://www.itl.nist.gov/div898/handbook/eda/section3/eda366d.htm (5 of 7) [5/1/2006 9:58:10 AM] Inverse Survival Function The formula for the inverse survival function of the power normal distribution is The following is the plot of the power normal inverse survival function with the same values of p as the pdf plots above. 1.3.6.6.13. Power Normal Distribution http://www.itl.nist.gov/div898/handbook/eda/section3/eda366d.htm (6 of 7) [5/1/2006 9:58:10 AM] Common Statistics The statistics for the power normal distribution are complicated and require tables. Nelson discusses the mean, median, mode, and standard deviation of the power normal distribution and provides references to the appropriate tables. Software Most general purpose statistical software programs do not support the probability functions for the power normal distribution. Dataplot does support them. 1.3.6.6.13. Power Normal Distribution http://www.itl.nist.gov/div898/handbook/eda/section3/eda366d.htm (7 of 7) [5/1/2006 9:58:10 AM] Cumulative Distribution Function The formula for the cumulative distribution function of the power lognormal distribution is where is the cumulative distribution function of the standard normal distribution. The following is the plot of the power lognormal cumulative distribution function with the same values of p as the pdf plots above. 1.3.6.6.14. Power Lognormal Distribution http://www.itl.nist.gov/div898/handbook/eda/section3/eda366e.htm (2 of 6) [5/1/2006 9:58:19 AM] Percent Point Function The formula for the percent point function of the power lognormal distribution is where is the percent point function of the standard normal distribution. The following is the plot of the power lognormal percent point function with the same values of p as the pdf plots above. 1.3.6.6.14. Power Lognormal Distribution http://www.itl.nist.gov/div898/handbook/eda/section3/eda366e.htm (3 of 6) [5/1/2006 9:58:19 AM] [...]... http://www.itl.nist.gov/div898 /handbook/ eda/section3/eda366g.htm (3 of 12) [5/1 /20 06 9:58 :22 AM] 1.3.6.6.16 Extreme Value Type I Distribution The formula for the cumulative distribution function of the Gumbel distribution (maximum) is The following is the plot of the Gumbel cumulative distribution function for the maximum case http://www.itl.nist.gov/div898 /handbook/ eda/section3/eda366g.htm (4 of 12) [5/1 /20 06 9:58 :22 AM]... standard Gumbel distribution (maximum) reduces to The following is the plot of the Gumbel probability density function for the maximum case http://www.itl.nist.gov/div898 /handbook/ eda/section3/eda366g.htm (2 of 12) [5/1 /20 06 9:58 :22 AM] 1.3.6.6.16 Extreme Value Type I Distribution Since the general form of probability functions can be expressed in terms of the standard distribution, all subsequent formulas... http://www.itl.nist.gov/div898 /handbook/ eda/section3/eda366e.htm (4 of 6) [5/1 /20 06 9:58:19 AM] 1.3.6.6. 14 Power Lognormal Distribution Survival Function The formula for the survival function of the power lognormal distribution is The following is the plot of the power lognormal survival function with the same values of p as the pdf plots above http://www.itl.nist.gov/div898 /handbook/ eda/section3/eda366e.htm (5 of 6) [5/1 /20 06... http://www.itl.nist.gov/div898 /handbook/ eda/section3/eda366g.htm (5 of 12) [5/1 /20 06 9:58 :22 AM] 1.3.6.6.16 Extreme Value Type I Distribution Hazard Function The formula for the hazard function of the Gumbel distribution (minimum) is The following is the plot of the Gumbel hazard function for the minimum case The formula for the hazard function of the Gumbel distribution http://www.itl.nist.gov/div898 /handbook/ eda/section3/eda366g.htm... purpose statistical software programs do not support the probability functions for the Tukey-Lambda distribution Dataplot does support them http://www.itl.nist.gov/div898 /handbook/ eda/section3/eda366f.htm (4 of 4) [5/1 /20 06 9:58 :20 AM] 1.3.6.6.16 Extreme Value Type I Distribution The general formula for the probability density function of the Gumbel (maximum) distribution is where is the location parameter... of as the pdf plots above The following is the plot of the Tukey-Lambda percent point function with the same values of as the pdf plots above http://www.itl.nist.gov/div898 /handbook/ eda/section3/eda366f.htm (2 of 4) [5/1 /20 06 9:58 :20 AM] 1.3.6.6.15 Tukey-Lambda Distribution Other Probability Functions The Tukey-Lambda distribution is typically used to identify an appropriate distribution (see the comments... value of at or near 0. 14, then the data can be modeled with a normal distribution Values of less than this imply a heavy-tailed distribution (with -1 approximating a Cauchy) That is, as the optimal value of goes from 0. 14 to -1, increasingly heavy tails are implied Similarly, as the optimal value of becomes greater than 0. 14, shorter tails are implied http://www.itl.nist.gov/div898 /handbook/ eda/section3/eda366f.htm... plot of the Gumbel hazard function for the minimum case The formula for the hazard function of the Gumbel distribution http://www.itl.nist.gov/div898 /handbook/ eda/section3/eda366g.htm (6 of 12) [5/1 /20 06 9:58 :22 AM] ... increasingly heavy tails are implied Similarly, as the optimal value of becomes greater than 0. 14, shorter tails are implied http://www.itl.nist.gov/div898 /handbook/ eda/section3/eda366f.htm (3 of 4) [5/1 /20 06 9:58 :20 AM] 1.3.6.6.15 Tukey-Lambda Distribution As the Tukey-Lambda distribution is a symmetric distribution, the use of the Tukey-Lambda PPCC plot to determine a reasonable distribution to model... http://www.itl.nist.gov/div898 /handbook/ eda/section3/eda366e.htm (5 of 6) [5/1 /20 06 9:58:19 AM] 1.3.6.6. 14 Power Lognormal Distribution Inverse Survival Function The formula for the inverse survival function of the power lognormal distribution is The following is the plot of the power lognormal inverse survival function with the same values of p as the pdf plots above Common Statistics The statistics for the power lognormal distribution are complicated . Extreme Value Type I Distribution http://www.itl.nist.gov/div898 /handbook/ eda/section3/eda366g.htm (2 of 12) [5/1 /20 06 9:58 :22 AM] Since the general form of probability functions can be expressed. Extreme Value Type I Distribution http://www.itl.nist.gov/div898 /handbook/ eda/section3/eda366g.htm (4 of 12) [5/1 /20 06 9:58 :22 AM] Percent Point Function The formula for the percent point function. above. 1.3.6.6.15. Tukey-Lambda Distribution http://www.itl.nist.gov/div898 /handbook/ eda/section3/eda366f.htm (2 of 4) [5/1 /20 06 9:58 :20 AM] Other Probability Functions The Tukey-Lambda distribution is