Common Probability Distributions

calendar Dec 12, 2024 · 10 min read · probability distributions statistics normal gaussian  ·
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Visual guide to common probability distributions with their properties and use cases.

Normal (Gaussian) Distribution

Symmetric bell curve - most common distribution in nature.

$$ f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} $$
  • Mean: $\mu$
  • Variance: $\sigma^2$
  • Symmetric around mean
  • 68-95-99.7 rule (1σ, 2σ, 3σ)
---
config:
  themeVariables:
    xyChart:
      plotColorPalette: "#ff0000, #0000ff, #00cc00"
---
xychart-beta
    title "Normal Distribution - Different Standard Deviations (μ=0)"
    x-axis "Value" [-240.0, -230.2, -220.4, -210.6, -200.8, -191.0, -181.2, -171.4, -161.6, -151.8, -142.0, -132.2, -122.4, -112.7, -102.9, -93.1, -83.3, -73.5, -63.7, -53.9, -44.1, -34.3, -24.5, -14.7, -4.9, 4.9, 14.7, 24.5, 34.3, 44.1, 53.9, 63.7, 73.5, 83.3, 93.1, 102.9, 112.7, 122.4, 132.2, 142.0, 151.8, 161.6, 171.4, 181.2, 191.0, 200.8, 210.6, 220.4, 230.2, 240.0]
    y-axis "Density" 0 --> 0.014
    line "σ=30" [0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0001, 0.0003, 0.0007, 0.0014, 0.0027, 0.0045, 0.0069, 0.0095, 0.0118, 0.0131, 0.0131, 0.0118, 0.0095, 0.0069, 0.0045, 0.0027, 0.0014, 0.0007, 0.0003, 0.0001, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000]
    line "σ=50" [0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0001, 0.0001, 0.0002, 0.0004, 0.0006, 0.0010, 0.0014, 0.0020, 0.0027, 0.0035, 0.0045, 0.0054, 0.0063, 0.0071, 0.0076, 0.0079, 0.0079, 0.0076, 0.0071, 0.0063, 0.0054, 0.0045, 0.0035, 0.0027, 0.0020, 0.0014, 0.0010, 0.0006, 0.0004, 0.0002, 0.0001, 0.0001, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000]
    line "σ=70" [0.0000, 0.0000, 0.0000, 0.0001, 0.0001, 0.0001, 0.0002, 0.0003, 0.0004, 0.0005, 0.0007, 0.0010, 0.0012, 0.0016, 0.0019, 0.0024, 0.0028, 0.0033, 0.0038, 0.0042, 0.0047, 0.0051, 0.0054, 0.0056, 0.0057, 0.0057, 0.0056, 0.0054, 0.0051, 0.0047, 0.0042, 0.0038, 0.0033, 0.0028, 0.0024, 0.0019, 0.0016, 0.0012, 0.0010, 0.0007, 0.0005, 0.0004, 0.0003, 0.0002, 0.0001, 0.0001, 0.0001, 0.0000, 0.0000, 0.0000]
 1from scipy import stats
 2import numpy as np
 3
 4# Generate samples
 5samples = np.random.normal(mu=0, sigma=1, size=1000)
 6
 7# PDF
 8x = np.linspace(-4, 4, 100)
 9pdf = stats.norm.pdf(x, loc=0, scale=1)
10
11# CDF
12cdf = stats.norm.cdf(x, loc=0, scale=1)

Binomial Distribution

Discrete - number of successes in $n$ independent trials.

$$ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} $$
  • Parameters: $n$ (trials), $p$ (success probability)
  • Mean: $np$
  • Variance: $np(1-p)$
---
config:
  themeVariables:
    xyChart:
      plotColorPalette: "#ff0000, #0000ff, #00cc00"
---
xychart-beta
    title "Binomial Distribution (n=20)"
    x-axis "Number of Successes" [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]
    y-axis "Probability" 0 --> 0.20
    line "p=0.3" [0.0008, 0.0068, 0.0278, 0.0716, 0.1304, 0.1789, 0.1916, 0.1643, 0.1144, 0.0654, 0.0308, 0.0120, 0.0039, 0.0010, 0.0002, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000]
    line "p=0.5" [0.0000, 0.0000, 0.0002, 0.0011, 0.0046, 0.0148, 0.0370, 0.0739, 0.1201, 0.1602, 0.1762, 0.1602, 0.1201, 0.0739, 0.0370, 0.0148, 0.0046, 0.0011, 0.0002, 0.0000, 0.0000]
    line "p=0.7" [0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0002, 0.0010, 0.0039, 0.0120, 0.0308, 0.0654, 0.1144, 0.1643, 0.1916, 0.1789, 0.1304, 0.0716, 0.0278, 0.0068, 0.0008, 0.0000]
1# Binomial: n=10 trials, p=0.5 success probability
2samples = np.random.binomial(n=10, p=0.5, size=1000)

Poisson Distribution

Discrete - number of events in fixed interval (rare events).

$$ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} $$
  • Parameter: $\lambda$ (average rate)
  • Mean: $\lambda$
  • Variance: $\lambda$
  • Models: arrivals, defects, calls per hour
---
config:
  themeVariables:
    xyChart:
      plotColorPalette: "#ff0000, #0000ff, #00cc00"
---
xychart-beta
    title "Poisson Distribution"
    x-axis "Number of Events" [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]
    y-axis "Probability" 0 --> 0.30
    line "λ=2" [0.1353, 0.2707, 0.2707, 0.1804, 0.0902, 0.0361, 0.0120, 0.0034, 0.0009, 0.0002, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000]
    line "λ=5" [0.0067, 0.0337, 0.0842, 0.1404, 0.1755, 0.1755, 0.1462, 0.1044, 0.0653, 0.0363, 0.0181, 0.0082, 0.0034, 0.0013, 0.0005, 0.0002, 0.0001, 0.0000, 0.0000, 0.0000, 0.0000]
    line "λ=10" [0.0000, 0.0005, 0.0023, 0.0076, 0.0189, 0.0378, 0.0631, 0.0901, 0.1126, 0.1251, 0.1251, 0.1137, 0.0948, 0.0729, 0.0521, 0.0347, 0.0217, 0.0128, 0.0071, 0.0037, 0.0019]
1# Poisson: lambda=3 average events
2samples = np.random.poisson(lam=3, size=1000)

Exponential Distribution

Continuous - time between events (memoryless property).

$$ f(x) = \lambda e^{-\lambda x}, \quad x \geq 0 $$
  • Parameter: $\lambda$ (rate)
  • Mean: $1/\lambda$
  • Variance: $1/\lambda^2$
  • Memoryless: $P(X > s+t | X > s) = P(X > t)$
---
config:
  themeVariables:
    xyChart:
      plotColorPalette: "#ff0000, #0000ff, #00cc00"
---
xychart-beta
    title "Exponential Distribution"
    x-axis "Time" [0.0, 0.2, 0.3, 0.5, 0.7, 0.8, 1.0, 1.1, 1.3, 1.5, 1.6, 1.8, 2.0, 2.1, 2.3, 2.4, 2.6, 2.8, 2.9, 3.1, 3.3, 3.4, 3.6, 3.8, 3.9, 4.1, 4.2, 4.4, 4.6, 4.7, 4.9, 5.1, 5.2, 5.4, 5.6, 5.7, 5.9, 6.0, 6.2, 6.4, 6.5, 6.7, 6.9, 7.0, 7.2, 7.3, 7.5, 7.7, 7.8, 8.0]
    y-axis "Density" 0 --> 2.0
    line "λ=0.5" [0.5000, 0.4608, 0.4247, 0.3914, 0.3607, 0.3324, 0.3064, 0.2824, 0.2602, 0.2398, 0.2210, 0.2037, 0.1877, 0.1730, 0.1595, 0.1470, 0.1354, 0.1248, 0.1150, 0.1060, 0.0977, 0.0900, 0.0830, 0.0765, 0.0705, 0.0650, 0.0599, 0.0552, 0.0509, 0.0469, 0.0432, 0.0398, 0.0367, 0.0338, 0.0312, 0.0287, 0.0265, 0.0244, 0.0225, 0.0207, 0.0191, 0.0176, 0.0162, 0.0149, 0.0138, 0.0127, 0.0117, 0.0108, 0.0099, 0.0092]
    line "λ=1.0" [1.0000, 0.8494, 0.7214, 0.6128, 0.5205, 0.4421, 0.3755, 0.3189, 0.2709, 0.2301, 0.1954, 0.1660, 0.1410, 0.1197, 0.1017, 0.0864, 0.0734, 0.0623, 0.0529, 0.0450, 0.0382, 0.0324, 0.0275, 0.0234, 0.0199, 0.0169, 0.0143, 0.0122, 0.0103, 0.0088, 0.0075, 0.0063, 0.0054, 0.0046, 0.0039, 0.0033, 0.0028, 0.0024, 0.0020, 0.0017, 0.0015, 0.0012, 0.0011, 0.0009, 0.0008, 0.0006, 0.0005, 0.0005, 0.0004, 0.0003]
    line "λ=2.0" [2.0000, 1.4428, 1.0409, 0.7509, 0.5417, 0.3908, 0.2819, 0.2034, 0.1467, 0.1059, 0.0764, 0.0551, 0.0397, 0.0287, 0.0207, 0.0149, 0.0108, 0.0078, 0.0056, 0.0040, 0.0029, 0.0021, 0.0015, 0.0011, 0.0008, 0.0006, 0.0004, 0.0003, 0.0002, 0.0002, 0.0001, 0.0001, 0.0001, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000]
1# Exponential: lambda=0.5
2samples = np.random.exponential(scale=1/0.5, size=1000)

Gamma Distribution

Continuous - sum of exponential random variables, waiting time for $k$ events.

$$ f(x) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x}, \quad x \geq 0 $$
  • Parameters: $\alpha$ (shape), $\beta$ (rate)
  • Mean: $\alpha/\beta$
  • Variance: $\alpha/\beta^2$
  • Special case: $\alpha=1$ gives Exponential
---
config:
  themeVariables:
    xyChart:
      plotColorPalette: "#ff0000, #0000ff, #00cc00, #cc00cc"
---
xychart-beta
    title "Gamma Distribution"
    x-axis "x" [0.0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0, 2.2, 2.4, 2.7, 2.9, 3.1, 3.3, 3.5, 3.7, 3.9, 4.1, 4.3, 4.5, 4.7, 4.9, 5.1, 5.3, 5.5, 5.7, 5.9, 6.1, 6.3, 6.5, 6.7, 6.9, 7.1, 7.3, 7.6, 7.8, 8.0, 8.2, 8.4, 8.6, 8.8, 9.0, 9.2, 9.4, 9.6, 9.8, 10.0]
    y-axis "Density" 0 --> 1.0
    line "α=1,β=1" [1.0000, 0.8154, 0.6649, 0.5421, 0.4421, 0.3604, 0.2939, 0.2397, 0.1954, 0.1593, 0.1299, 0.1059, 0.0864, 0.0704, 0.0574, 0.0468, 0.0382, 0.0311, 0.0254, 0.0207, 0.0169, 0.0138, 0.0112, 0.0092, 0.0075, 0.0061, 0.0050, 0.0040, 0.0033, 0.0027, 0.0022, 0.0018, 0.0015, 0.0012, 0.0010, 0.0008, 0.0006, 0.0005, 0.0004, 0.0003, 0.0003, 0.0002, 0.0002, 0.0002, 0.0001, 0.0001, 0.0001, 0.0001, 0.0001, 0.0000]
    line "α=2,β=1" [0.0000, 0.1664, 0.2714, 0.3319, 0.3609, 0.3678, 0.3599, 0.3424, 0.3190, 0.2927, 0.2651, 0.2378, 0.2115, 0.1869, 0.1641, 0.1434, 0.1247, 0.1080, 0.0933, 0.0803, 0.0689, 0.0590, 0.0504, 0.0430, 0.0365, 0.0310, 0.0263, 0.0223, 0.0188, 0.0159, 0.0134, 0.0113, 0.0095, 0.0080, 0.0067, 0.0056, 0.0047, 0.0040, 0.0033, 0.0028, 0.0023, 0.0019, 0.0016, 0.0014, 0.0011, 0.0009, 0.0008, 0.0007, 0.0005, 0.0005]
    line "α=3,β=1" [0.0000, 0.0170, 0.0554, 0.1016, 0.1473, 0.1877, 0.2203, 0.2445, 0.2604, 0.2688, 0.2706, 0.2669, 0.2590, 0.2479, 0.2344, 0.2194, 0.2036, 0.1874, 0.1713, 0.1556, 0.1406, 0.1264, 0.1131, 0.1008, 0.0895, 0.0792, 0.0698, 0.0614, 0.0539, 0.0471, 0.0411, 0.0358, 0.0311, 0.0270, 0.0233, 0.0202, 0.0174, 0.0150, 0.0129, 0.0111, 0.0095, 0.0081, 0.0070, 0.0059, 0.0051, 0.0043, 0.0037, 0.0031, 0.0027, 0.0023]
    line "α=5,β=1" [0.0000, 0.0001, 0.0008, 0.0032, 0.0082, 0.0163, 0.0275, 0.0416, 0.0579, 0.0756, 0.0939, 0.1121, 0.1295, 0.1454, 0.1595, 0.1714, 0.1809, 0.1880, 0.1926, 0.1950, 0.1952, 0.1935, 0.1900, 0.1851, 0.1789, 0.1718, 0.1639, 0.1554, 0.1465, 0.1375, 0.1284, 0.1194, 0.1105, 0.1019, 0.0936, 0.0857, 0.0782, 0.0712, 0.0646, 0.0584, 0.0527, 0.0475, 0.0426, 0.0382, 0.0341, 0.0304, 0.0271, 0.0241, 0.0214, 0.0189]
1from scipy import stats
2
3# Gamma distribution
4alpha, beta = 2, 1
5samples = np.random.gamma(alpha, 1/beta, size=1000)
6
7# Or using scipy
8x = np.linspace(0, 10, 100)
9pdf = stats.gamma.pdf(x, a=alpha, scale=1/beta)

Pareto Distribution

Heavy-tailed - power law distribution (80/20 rule, wealth distribution).

$$ f(x) = \frac{\alpha x_m^\alpha}{x^{\alpha+1}}, \quad x \geq x_m $$
  • Parameters: $\alpha$ (shape), $x_m$ (scale/minimum)
  • Mean: $\frac{\alpha x_m}{\alpha - 1}$ for $\alpha > 1$
  • Heavy tail: $P(X > x) \propto x^{-\alpha}$
---
config:
  themeVariables:
    xyChart:
      plotColorPalette: "#ff0000, #0000ff, #00cc00, #cc00cc"
---
xychart-beta
    title "Pareto Distribution (Heavy Tail)"
    x-axis "x" [1.0, 1.1, 1.3, 1.4, 1.6, 1.7, 1.9, 2.0, 2.1, 2.3, 2.4, 2.6, 2.7, 2.9, 3.0, 3.1, 3.3, 3.4, 3.6, 3.7, 3.9, 4.0, 4.1, 4.3, 4.4, 4.6, 4.7, 4.9, 5.0, 5.1, 5.3, 5.4, 5.6, 5.7, 5.9, 6.0, 6.1, 6.3, 6.4, 6.6, 6.7, 6.9, 7.0, 7.1, 7.3, 7.4, 7.6, 7.7, 7.9, 8.0]
    y-axis "Density" 0 --> 3.0
    line "α=1.5" [1.5000, 1.0743, 0.8003, 0.6149, 0.4846, 0.3898, 0.3191, 0.2652, 0.2232, 0.1899, 0.1632, 0.1415, 0.1236, 0.1087, 0.0962, 0.0857, 0.0767, 0.0689, 0.0622, 0.0564, 0.0513, 0.0469, 0.0429, 0.0394, 0.0363, 0.0336, 0.0311, 0.0288, 0.0268, 0.0250, 0.0234, 0.0218, 0.0205, 0.0192, 0.0181, 0.0170, 0.0160, 0.0151, 0.0143, 0.0136, 0.0128, 0.0122, 0.0116, 0.0110, 0.0105, 0.0100, 0.0095, 0.0091, 0.0087, 0.0083]
    line "α=2.0" [2.0000, 1.3398, 0.9410, 0.6860, 0.5154, 0.3970, 0.3122, 0.2500, 0.2033, 0.1675, 0.1396, 0.1176, 0.1000, 0.0858, 0.0741, 0.0644, 0.0564, 0.0496, 0.0439, 0.0390, 0.0349, 0.0312, 0.0281, 0.0254, 0.0230, 0.0209, 0.0191, 0.0175, 0.0160, 0.0147, 0.0135, 0.0125, 0.0116, 0.0107, 0.0100, 0.0093, 0.0086, 0.0081, 0.0075, 0.0070, 0.0066, 0.0062, 0.0058, 0.0055, 0.0052, 0.0049, 0.0046, 0.0044, 0.0041, 0.0039]
    line "α=3.0" [3.0000, 1.7585, 1.0979, 0.7203, 0.4920, 0.3474, 0.2522, 0.1875, 0.1423, 0.1099, 0.0862, 0.0686, 0.0553, 0.0450, 0.0370, 0.0307, 0.0257, 0.0217, 0.0184, 0.0158, 0.0136, 0.0117, 0.0102, 0.0089, 0.0078, 0.0069, 0.0061, 0.0054, 0.0048, 0.0043, 0.0038, 0.0035, 0.0031, 0.0028, 0.0025, 0.0023, 0.0021, 0.0019, 0.0018, 0.0016, 0.0015, 0.0014, 0.0012, 0.0012, 0.0011, 0.0010, 0.0009, 0.0008, 0.0008, 0.0007]
    line "α=5.0" [5.0000, 2.2440, 1.1069, 0.5882, 0.3320, 0.1970, 0.1219, 0.0781, 0.0516, 0.0351, 0.0244, 0.0173, 0.0125, 0.0092, 0.0069, 0.0052, 0.0040, 0.0031, 0.0024, 0.0019, 0.0015, 0.0012, 0.0010, 0.0008, 0.0007, 0.0005, 0.0005, 0.0004, 0.0003, 0.0003, 0.0002, 0.0002, 0.0002, 0.0001, 0.0001, 0.0001, 0.0001, 0.0001, 0.0001, 0.0001, 0.0001, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000]

Note: Heavy tail means rare extreme events are more likely than in normal distribution.

1from scipy import stats
2
3# Pareto distribution
4alpha, xm = 2, 1
5samples = (np.random.pareto(alpha, size=1000) + 1) * xm
6
7# Or using scipy
8x = np.linspace(xm, 10, 100)
9pdf = stats.pareto.pdf(x, alpha, scale=xm)

Skewed Distributions

Log-Normal Distribution

Right-skewed - exponential of normal distribution.

$$ f(x) = \frac{1}{x\sigma\sqrt{2\pi}} e^{-\frac{(\ln x - \mu)^2}{2\sigma^2}}, \quad x > 0 $$
---
config:
  themeVariables:
    xyChart:
      plotColorPalette: "#ff0000, #0000ff, #00cc00"
---
xychart-beta
    title "Log-Normal Distribution (Right-Skewed)"
    x-axis "x" [0.1, 0.3, 0.4, 0.6, 0.7, 0.9, 1.1, 1.2, 1.4, 1.6, 1.7, 1.9, 2.0, 2.2, 2.4, 2.5, 2.7, 2.8, 3.0, 3.2, 3.3, 3.5, 3.6, 3.8, 4.0, 4.1, 4.3, 4.5, 4.6, 4.8, 4.9, 5.1, 5.3, 5.4, 5.6, 5.7, 5.9, 6.1, 6.2, 6.4, 6.5, 6.7, 6.9, 7.0, 7.2, 7.4, 7.5, 7.7, 7.8, 8.0]
    y-axis "Density" 0 --> 1.0
    line "μ=0,σ=0.5" [0.0002, 0.0831, 0.4278, 0.7656, 0.9005, 0.8636, 0.7412, 0.5967, 0.4623, 0.3499, 0.2613, 0.1936, 0.1429, 0.1054, 0.0778, 0.0575, 0.0427, 0.0317, 0.0237, 0.0178, 0.0134, 0.0101, 0.0077, 0.0059, 0.0045, 0.0035, 0.0027, 0.0021, 0.0016, 0.0013, 0.0010, 0.0008, 0.0006, 0.0005, 0.0004, 0.0003, 0.0002, 0.0002, 0.0002, 0.0001, 0.0001, 0.0001, 0.0001, 0.0001, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000]
    line "μ=0,σ=1.0" [0.2816, 0.6203, 0.6515, 0.5913, 0.5128, 0.4381, 0.3730, 0.3179, 0.2719, 0.2336, 0.2016, 0.1749, 0.1523, 0.1333, 0.1172, 0.1034, 0.0916, 0.0814, 0.0726, 0.0650, 0.0583, 0.0525, 0.0474, 0.0428, 0.0389, 0.0353, 0.0322, 0.0294, 0.0269, 0.0246, 0.0226, 0.0208, 0.0191, 0.0176, 0.0163, 0.0151, 0.0140, 0.0130, 0.0120, 0.0112, 0.0104, 0.0097, 0.0091, 0.0085, 0.0079, 0.0074, 0.0069, 0.0065, 0.0061, 0.0057]
    line "μ=0,σ=1.5" [0.8187, 0.6822, 0.5338, 0.4272, 0.3502, 0.2929, 0.2489, 0.2145, 0.1868, 0.1643, 0.1457, 0.1301, 0.1168, 0.1056, 0.0958, 0.0874, 0.0800, 0.0735, 0.0677, 0.0626, 0.0581, 0.0540, 0.0503, 0.0469, 0.0439, 0.0412, 0.0387, 0.0364, 0.0343, 0.0324, 0.0306, 0.0289, 0.0274, 0.0260, 0.0247, 0.0235, 0.0224, 0.0213, 0.0203, 0.0194, 0.0185, 0.0177, 0.0170, 0.0162, 0.0156, 0.0149, 0.0143, 0.0138, 0.0132, 0.0127]
1# Log-normal distribution
2mu, sigma = 0, 1
3samples = np.random.lognormal(mu, sigma, size=1000)

Distribution Comparison

DistributionTypeParametersMeanUse Case
NormalContinuous$\mu, \sigma$$\mu$Natural phenomena, errors
BinomialDiscrete$n, p$$np$Success/failure trials
PoissonDiscrete$\lambda$$\lambda$Rare events, arrivals
ExponentialContinuous$\lambda$$1/\lambda$Time between events
GammaContinuous$\alpha, \beta$$\alpha/\beta$Waiting times
ParetoContinuous$\alpha, x_m$$\frac{\alpha x_m}{\alpha-1}$Power laws, wealth
Log-NormalContinuous$\mu, \sigma$$e^{\mu + \sigma^2/2}$Multiplicative processes

Further Reading

Related Snippets