# 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:

• 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 - 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:

• 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