IEG 3050 tutorial 6 Input random variable

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IEG 3050 tutorial 6Input random variable

• Leo

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Content

• General idea
• Distribution
• Useful Summary Statistics
• Maximum Likelihood Estimation
• ?2 Goodness-of-Fit Test
• Q-Q Plot P-P Plot
• Conclusion

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General Idea

• After collecting data on an random variable of
  interest, these data can be used to specify a
  distribution
• Trace Driven
• Empirical distribution approach
• Fitted standard distribution approach

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General Idea
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General Idea

• Procedure for Fitted standard distribution
  approach

Step 1 select a distribution family -empirical
data -theoretical distribution
Step 2 parameter estimation -method of moment
(mean, variance) -method of maximum likelihood
estimator (MLE)
Step 3 determine how representative the fitted
distribution are -heuristic method -chi-square
of fit test -graphical aid (Q-Q plot, P-P plot)
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General Idea

• Procedure for Fitted standard distribution
  approach

Step 1 select a distribution family -empirical
data -theoretical distribution
3rd Way Summary Statistic Mean, median,
variance Coefficient of variation Lexis
ratio Skewness quantile summaries
1st Way Prior knowledge, physical reasoning
2nd Way By investigating histogram
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Distribution
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Distribution
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Distribution
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Distribution
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Useful Summary Statistics

• Procedure for Fitted standard distribution
  approach

Step 1 select a distribution family -empirical
data -theoretical distribution
3rd Way Summary Statistic Mean, median,
variance Coefficient of variation Lexis
ratio Skewness quantile summaries
1st Way Prior knowledge, physical reasoning
2nd Way By investigating histogram
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Useful Summary Statistics

• Mean, Median, Variance
• Coefficient of variation

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Useful Summary Statistics

• Lexis ration (for discrete r.v.)

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Useful Summary Statistics

• Skewness

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MLE

• Procedure for Fitted standard distribution
  approach

Step 1 select a distribution family -empirical
data -theoretical distribution
Step 2 parameter estimation -method of moment
(mean, variance) -method of maximum likelihood
estimator (MLE)
Step 3 determine how representative the fitted
distribution are -heuristic method -chi-square
of fit test -graphical aid (Q-Q plot, P-P plot)
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MLE
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MLE

• Exercise 1
• Suppose we have n(25) sample data x1 x2
  xn
• 1.77 0.27 0.64 3.53 1.27 1.39 0.19 1.00 1.96
  0.37 0.99 0.04 0.44 2.57 0.78 2.26 0.10 0.17 0.86
  0.51 0.48 1.22 0.15 0.49 0.81, you have assumed
  they are iid and exponential distributed, find ?

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MLE

• Exercise 1

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General Idea

• Procedure for Fitted standard distribution
  approach

Step 1 select a distribution family -empirical
data -theoretical distribution
Step 2 parameter estimation -method of moment
(mean, variance) -method of maximum likelihood
estimator (MLE)
Step 3 determine how representative the fitted
distribution are -heuristic method -chi-square
of fit test -graphical aid (Q-Q plot, P-P plot)
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Chi-square goodness of fit test
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Chi-square goodness of fit test
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Q-Q plot

• Procedure for Fitted standard distribution
  approach

Step 1 select a distribution family -empirical
data -theoretical distribution
Step 2 parameter estimation -method of moment
(mean, variance) -method of maximum likelihood
estimator (MLE)
Step 3 determine how representative the fitted
distribution are -heuristic method -chi-square
of fit test -graphical aid (Q-Q plot, P-P plot)
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Q-Q plot
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P-P plot
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Conclusion

• The idea of how to choose a suitable input random
  variable has been discussed
• Q/A