Probability distributions

by Nikolai V. Shokhirev

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In this section there are collected several distributions representing various magnitude of statistical parameters (Mean, Variance, Skewness and Kurtosis).

Uniform distribution

 Mean    0   
 Variance 
Skewness  0
Kurtosis *)  -6/5 = -1.2 

*) Here and below Kurtosis = Excess kurtosis. 

Triangular distribution

 Mean    (L2 - L1)/3  
 Variance   (L22 + L2 L1 + L12)/18 
Skewness  
Kurtosis  -3/5 = -0.6

Special cases

  L1 = L2  = L/2   L1 = 0,  L2  = L
 Mean    0  L /3
 Variance   L2 /24   L2 /18 
Skewness  0   = 0.565685
Kurtosis   -0.6    -0.6

The Pearson type VII distribution

 Mean    μ  
 Variance   a2 /(2 m - 3) , m > 3/2 
Skewness  0
Kurtosis  6/(2 m - 5) , m > 5/2

Special cases:

Normal (Gaussian) distribution

 Mean    μ
 Variance   σ2  
Skewness  0
Kurtosis  0

Normal - Pearson5/2 comparison


Comparison of the distributions with Variance = 1:
Red curve - Pearson5/2, Blue curve - Gaussian.

It hard to see the "fat" Pearson distribution tails, but its sharp peak is definitely noticeable.

Two-exponential distribution

 Mean    μ + L2 - L1
 Variance   L22 +  L12
Skewness  2( L23 -  L13)/( L22 +  L12)3/2  
Kurtosis  6( L24 +  L14)/( L22 +  L12)2  


L1 = 1, L2 = 2

Special Cases:

   L1 = 0, L2 = L   L1 =  L2 = L /2 
 Mean   μ + L   μ
 Variance   L2  2 L2
Skewness  2   0
Kurtosis  6    3

Power-exponential distribution

 Mean    2 L  x/L2 exp(-x/L) 

 0 < x < ∞
 Variance   2 L2
Skewness  
Kurtosis  3

Power-exponential distribution for L = 1

Raised cosine distribution

 Mean    μ

  μ - s < x <  μ + s

 Variance   
 = s2 0.13069096604865776
Skewness  0
Kurtosis  1.2 (90 - π4)/(π2 - 6)2 =
  -05937628756


s = 1, μ = 0

Wigner semicircle distribution

 

 Mean    0  2 (R2 - x2)1/2/(π R2)  

 -R < x < R

 Variance   R2/4
Skewness  0
Kurtosis  -1
In general, this is the upper half of an ellipse.


R = 1

References

  1. http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29
  2. http://en.wikipedia.org/wiki/Triangular_distribution
  3. http://en.wikipedia.org/wiki/Pearson_distribution
  4. http://en.wikipedia.org/wiki/Normal_distribution
  5. http://en.wikipedia.org/wiki/Raised_cosine_distribution
  6. PROB - Probability Density Functions http://orion.math.iastate.edu/burkardt/f_src/prob/prob.html

Tools used

  1. Notepad
  2. Mathematica 4
  3. Scientific calculator by Nikolai Shokhirev
  4. Paint Shop Pro 5

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Up: ABC Stat
- Definitions
- Distributions
- Multivariate correlations
- Principal Component Analysis
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