Introduction

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:

• m → 5/2  Kurtosis → ∞ (  p(x) = 3 [2 + x²]^-5/2 , see the plot below )
• m → ∞  Kurtosis → 0 (Normal distribution)

Normal (Gaussian) distribution

Mean	μ
Variance	σ2
Skewness	0
Kurtosis	0

Normal - Pearson_5/2 comparison

Comparison of the distributions with Variance = 1:
Red curve - Pearson_5/2, 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:

• Case L₁ = 0, L₂ = L - exponential distribution
• Case L₁ = L₂ = L - Laplace distribution

	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