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Summary Probability Distributions
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  • Course
  • Institution
  • Islamitische Universiteit Rotterdam (IUASR)
  • Book
  • John E. Freund\'s Mathematical Statistics with Applications: Pearson International Edition

This is a summary for the subject Probability Distributions of the Econometrics and Operations Research study. The summary is from the Chapters 6, 7, 8 and 10 of the book Mathematical Statistics with applications by John E. Freund's.

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Document information

  • No
  • H6, h7, h8, h10
  • June 5, 2017
  • 14
  • 2016/2017
  • Summary

Subjects

  • econometrie
  • econometrics
  • probability distributions
  • mathematical
  • statistics

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  • Islamitische Universiteit Rotterdam (IUASR)
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Chapter 6

Uniform Distribution
Probability density is given by:
1
𝑓𝑜𝑟 𝛼 < 𝑥 < 𝛽
𝑢(𝑥; 𝛼, 𝛽) = {𝛽 − 𝛼
0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
Where 𝛼, 𝛽 are real constants with 𝛼 < 𝛽.
The mean and the variance:
𝛼+𝛽 1
𝜇= 𝑎𝑛𝑑 𝜎2 = (𝛽 − 𝛼)2
2 12

Gamma Distribution
Probability density is given by:
1 𝑥
𝛼−1 𝑒 −𝛽
𝑥 𝑓𝑜𝑟 𝑥 > 0
𝑔(𝑥; 𝛼, 𝛽) = { 𝛽 𝛼 𝛤(𝛼)
0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
Where 𝛼, 𝛽 > 0.
The mean and the variance:
𝜇 = 𝛼𝛽 𝑎𝑛𝑑 𝜎 2 = 𝛼𝛽2
Some notes:
1. Gamma function:
∞
𝛤(𝛼) = ∫0 𝑦 𝛼−1 𝑒 −𝑦 𝑑𝑦 for 𝛼 > 0
2. 𝛤(𝛼) = (𝛼 − 1) ∙ 𝛤(𝛼 − 1) for 𝛼 > 1
Prove: using integration by parts:
∞ ∞
∫0 𝑢𝑣 ′ 𝑑𝑥 = 𝑢𝑣 − ∫0 𝑢′𝑣 𝑑𝑥 (where u is the simplest function)
3. 𝛤(1) = 1
4. 𝛤(𝛼) = ( 𝛼 − 1)! When 𝛼 is a positive integer
1
5. 𝛤 (2) = √𝜋
When 𝛼 is not a positive integer, the value of 𝛤(𝛼) will have to be looked up in a special table.

Exponential Distribution
Probability density is given by:
1 −𝑥
𝑔(𝑥; 𝜃) = { 𝜃 𝑒 𝑓𝑜𝑟 𝑥 > 0
𝜃

0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
Where 𝜃 > 0.
This is the special case of the Gamma Distribution with 𝛼 = 1 and 𝛽 = 𝜃.
1
Using 𝜃 = we can obtain the waiting time between successes.
𝜆
The mean and the variance:
𝜇= 𝜃 𝑎𝑛𝑑 𝜎2 = 𝜃2



The exponential distribution applies to:
- The occurrence of the first success in a Poisson process
- The waiting time between two successes

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