Continuous Distribution Generators"

In RandomWalker: Generate Random Walks Compatible with the 'tidyverse'

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  fig.width = 7,
  fig.height = 5
)

library(RandomWalker)
library(dplyr)
library(ggplot2)
library(patchwork)

RandomWalker provides a comprehensive suite of random walk generators based on continuous probability distributions. Each function follows a consistent API design for ease of use.

Table of Contents

• Common Parameters
• Normal Distribution
• Brownian Motion
• Geometric Brownian Motion
• Beta Distribution
• Cauchy Distribution
• Chi-Squared Distribution
• Exponential Distribution
• F Distribution
• Gamma Distribution
• Log-Normal Distribution
• Logistic Distribution
• Student's t Distribution
• Uniform Distribution
• Weibull Distribution
• Comparison Guide

Common Parameters

All continuous distribution generators share these common parameters:

| Parameter | Type | Description | Default | |-----------|------|-------------|---------| | .num_walks | Integer | Number of walks to generate | 25 | | .n | Integer | Number of steps per walk | 100 | | .initial_value | Numeric | Starting value for the walk | 0 | | .dimensions | Integer | Spatial dimensions (1, 2, or 3) | 1 |

Additional parameters are distribution-specific and control the shape of the probability distribution.

Normal Distribution

random_normal_walk()

The most commonly used random walk, based on the normal (Gaussian) distribution.

Function Signature:

random_normal_walk(
  .num_walks = 25,
  .n = 100,
  .mu = 0,
  .sd = 1,
  .initial_value = 0,
  .dimensions = 1
)

Distribution Parameters: - .mu - Mean of the distribution (default: 0) - .sd - Standard deviation (default: 1)

Use Cases: - General-purpose random walks - Modeling measurement errors - Simulating natural phenomena with normal noise

Example:

# Basic normal walk
random_normal_walk(.num_walks = 10, .n = 100) |>
  visualize_walks()

# With custom mean and SD
random_normal_walk(
  .num_walks = 5,
  .n = 200,
  .mu = 0.01,    # Slight upward drift
  .sd = 0.5,     # Lower volatility
  .initial_value = 100
) |> visualize_walks()

# 2D spatial walk
random_normal_walk(
  .num_walks = 3,
  .n = 500,
  .dimensions = 2
) |> visualize_walks()

random_normal_drift_walk()

Random walk with deterministic drift component.

Function Signature:

random_normal_drift_walk(
  .num_walks = 25,
  .n = 100,
  .mu = 0,
  .sd = 1,
  .initial_value = 0,
  .dimensions = 1
)

Key Difference from random_normal_walk(): - .drift - Drift term added to each step (default: 0.1) - Adds explicit drift term: drift + random_step - More pronounced trending behavior - Better for modeling processes with clear directional movement

Example:

# Compare walks with and without drift
p1 <- random_normal_walk(.num_walks = 5, .mu = 0.1) |>
  visualize_walks(.pluck = "y") +
  labs(title = "Normal Walk")

p2 <- random_normal_drift_walk(.num_walks = 5, .mu = 0.1) |>
  visualize_walks(.pluck = "y") +
  labs(title = "Normal Walk with Drift")

p1 / p2

Brownian Motion

brownian_motion()

Standard Brownian motion (Wiener process) - the foundation of stochastic calculus.

Function Signature:

brownian_motion(
  .num_walks = 25,
  .n = 100,
  .delta_time = 1,
  .initial_value = 0,
  .dimensions = 1
)

Parameters: - .num_walks - Number of walks to generate (default: 25) - .n - Number of steps per walk (default: 100) - .delta_time - Time increment per step (default: 1) - .initial_value - Starting value for each walk (default: 0) - .dimensions - Number of dimensions (default: 1)

Mathematical Form:

Use Cases: - Financial mathematics (Black-Scholes model) - Physics (particle diffusion) - Signal processing (noise modeling)

Example:

# Standard Brownian motion
brownian_motion(.num_walks = 10) |>
  visualize_walks()

# With drift and volatility
brownian_motion(
  .num_walks = 50,
  .n = 252,
  .delta_time = 0.05,
  .initial_value = 100
) |> visualize_walks(.alpha = 0.3)

geometric_brownian_motion()

Geometric Brownian motion - the standard model for stock prices.

Function Signature:

geometric_brownian_motion(
  .num_walks = 25,
  .n = 100,
  .mu = 0,
  .sigma = 0.1,
  .initial_value = 100,
  .delta_time = 0.003,
  .dimensions = 1
)

Parameters: - .mu - Expected return (drift) - .sigma - Volatility

Mathematical Form:

dS(t) = μ S(t) dt + σ S(t) dW(t)
S(t) = S(0) exp((μ - σ²/2)t + σW(t))

Key Properties: - Always positive (can't go below zero) - Log-normal distribution of prices - Percentage changes are normally distributed

Use Cases: - Stock price modeling - Option pricing - Asset allocation simulations - Monte Carlo risk analysis

Example:

# Model stock prices
stock_sim <- geometric_brownian_motion(
  .num_walks = 1000,
  .n = 252,        # Trading days in a year
  .mu = 0.08,      # 8% expected return
  .sigma = 0.25,   # 25% volatility
  .initial_value = 100
)

# Visualize scenarios
stock_sim |> visualize_walks(.alpha = 0.1)

# Analyze outcomes
stock_sim |>
  summarize_walks(.value = cum_prod_y, .group_var = walk_number) |>
  summarize(
    median_price = median(max_val),
    prob_profit = mean(max_val > 100),
    prob_loss_50 = mean(min_val < 50)
  )

Beta Distribution

random_beta_walk()

Random walk based on the beta distribution (bounded between 0 and 1).

Function Signature:

random_beta_walk(
  .num_walks = 25,
  .n = 100,
  .shape1 = 2,
  .shape2 = 2,
  .ncp = 0,
  .initial_value = 0,
  .samp = TRUE,
  .replace = TRUE,
  .sample_size = 0.8,
  .dimensions = 1
)

Parameters: - .shape1 - First shape parameter (α) - .shape2 - Second shape parameter (β) - .ncp - Non-centrality parameter (default: 0)

Properties: - Steps are bounded: 0 ≤ step ≤ 1 - Flexible shapes (uniform, U-shaped, bell-shaped) - Mean = α / (α + β) - Variance = (α β) / ((α + β)² (α + β + 1))

Shape Guide: - shape1 = 1, shape2 = 1: Uniform (0,1) - shape1 = 2, shape2 = 2: Symmetric, bell-shaped - shape1 = 2, shape2 = 5: Right-skewed - shape1 = 5, shape2 = 2: Left-skewed

Use Cases: - Modeling proportions or percentages - Bounded processes (e.g., utilization rates) - Success rates over time

Example:

# Symmetric beta walk
random_beta_walk(
  .num_walks = 10,
  .shape1 = 2,
  .shape2 = 2
) |> visualize_walks()

# Right-skewed (toward 0)
random_beta_walk(
  .num_walks = 10,
  .shape1 = 2,
  .shape2 = 5
) |> visualize_walks()

# Compare different shapes
p1 <- random_beta_walk(.shape1 = 1, .shape2 = 1) |>
  visualize_walks(.pluck = "y")
p2 <- random_beta_walk(.shape1 = 2, .shape2 = 5) |>
  visualize_walks(.pluck = "y")
p1 / p2

Cauchy Distribution

random_cauchy_walk()

Random walk with heavy tails - extreme values are much more common than in normal distribution.

Function Signature:

random_cauchy_walk(
  .num_walks = 25,
  .n = 100,
  .location = 0,
  .scale = 1,
  .initial_value = 0,
  .samp = TRUE,
  .replace = TRUE,
  .sample_size = 0.8,
  .dimensions = 1
)

Parameters: - .location - Location parameter (median) - .scale - Scale parameter (spread)

Properties: - No defined mean or variance! - Heavy tails (high probability of extreme values) - Symmetric around location - Much more volatile than normal distribution

Use Cases: - Modeling extreme events - Financial crises simulation - Physics (resonance phenomena) - Robust statistics demonstrations

Example:

# Standard Cauchy walk
random_cauchy_walk(.num_walks = 10) |>
  visualize_walks()

# Compare with normal walk
p1 <- random_normal_walk(.num_walks = 5, .sd = 1) |>
  visualize_walks(.pluck = "y") +
  labs(title = "Normal Walk")

p2 <- random_cauchy_walk(.num_walks = 5, .scale = 1) |>
  visualize_walks(.pluck = "y") +
  labs(title = "Cauchy Walk (Heavy Tails)")

p1 / p2

Chi-Squared Distribution

random_chisquared_walk()

Random walk based on chi-squared distribution (always positive).

Function Signature:

random_chisquared_walk(
  .num_walks = 25,
  .n = 100,
  .df = 5,
  .ncp = 0,
  .initial_value = 0,
  .samp = TRUE,
  .replace = TRUE,
  .sample_size = 0.8,
  .dimensions = 1
)

Parameters: - .df - Degrees of freedom - .ncp - Non-centrality parameter (default: 0)

Properties: - Steps are always positive - Right-skewed (especially for low df) - Mean = df - Variance = 2df

Use Cases: - Goodness-of-fit tests - Variance estimation - Quality control - Reliability engineering

Example:

# Low df (very skewed)
random_chisquared_walk(.num_walks = 10, .df = 1) |>
  visualize_walks()

# Higher df (more symmetric)
random_chisquared_walk(.num_walks = 10, .df = 10) |>
  visualize_walks()

Exponential Distribution

random_exponential_walk()

Random walk with exponentially distributed steps (memoryless property).

Function Signature:

random_exponential_walk(
  .num_walks = 25,
  .n = 100,
  .rate = 1,
  .initial_value = 0,
  .samp = TRUE,
  .replace = TRUE,
  .sample_size = 0.8,
  .dimensions = 1
)

Parameters: - .rate - Rate parameter (λ)

Properties: - Steps are always positive - Memoryless property - Mean = 1/rate - Variance = 1/rate²

Use Cases: - Time between events (queuing theory) - Lifetime modeling - Radioactive decay - Poisson process intervals

Example:

# Standard exponential
random_exponential_walk(.num_walks = 10) |>
  visualize_walks()

# Fast rate (smaller steps)
random_exponential_walk(.num_walks = 10, .rate = 5) |>
  visualize_walks()

# Slow rate (larger steps)
random_exponential_walk(.num_walks = 10, .rate = 0.5) |>
  visualize_walks()

F Distribution

random_f_walk()

Random walk based on F-distribution (ratio of chi-squared variables).

Function Signature:

random_f_walk(
  .num_walks = 25,
  .n = 100,
  .df1 = 5,
  .df2 = 5,
  .ncp = NULL,
  .initial_value = 0,
  .samp = TRUE,
  .replace = TRUE,
  .sample_size = 0.8,
  .dimensions = 1
)

Parameters: - .df1 - Numerator degrees of freedom - .df2 - Denominator degrees of freedom - .ncp - Non-centrality parameter

Use Cases: - ANOVA - Variance ratio tests - Model comparison

Example:

random_f_walk(.num_walks = 10, .df1 = 5, .df2 = 10) |>
  visualize_walks()

Gamma Distribution

random_gamma_walk()

Flexible distribution for positive values.

Function Signature:

random_gamma_walk(
  .num_walks = 25,
  .n = 100,
  .shape = 1,
  .scale = 1,
  .rate = NULL,
  .initial_value = 0,
  .samp = TRUE,
  .replace = TRUE,
  .sample_size = 0.8,
  .dimensions = 1
)

Parameters: - .shape - Shape parameter (k) - .rate - Rate parameter (β) - .scale - Scale parameter (1/rate)

Properties: - Steps always positive - Flexible shapes - Mean = shape/rate - Variance = shape/rate²

Use Cases: - Waiting times - Rainfall modeling - Insurance claims - Reliability analysis

Example:

# Different shape parameters
random_gamma_walk(.num_walks = 10, .shape = 1, .rate = 1) |>
  visualize_walks()

random_gamma_walk(.num_walks = 10, .shape = 5, .rate = 1) |>
  visualize_walks()

Log-Normal Distribution

random_lognormal_walk()

Random walk where log of steps is normally distributed.

Function Signature:

random_lognormal_walk(
  .num_walks = 25,
  .n = 100,
  .meanlog = 0,
  .sdlog = 1,
  .initial_value = 0,
  .samp = TRUE,
  .replace = TRUE,
  .sample_size = 0.8,
  .dimensions = 1
)

Parameters: - .meanlog - Mean of log - .sdlog - Standard deviation of log

Properties: - Steps always positive - Right-skewed - Multiplicative processes

Use Cases: - Income distributions - File sizes - Particle sizes - Stock prices (alternative to GBM)

Example:

random_lognormal_walk(.num_walks = 10) |>
  visualize_walks()

Logistic Distribution

random_logistic_walk()

Random walk with logistic distribution (heavier tails than normal).

Function Signature:

random_logistic_walk(
  .num_walks = 25,
  .n = 100,
  .location = 0,
  .scale = 1,
  .initial_value = 0,
  .samp = TRUE,
  .replace = TRUE,
  .sample_size = 0.8,
  .dimensions = 1
)

Use Cases: - Growth models - Binary classification - Neural networks

Example:

random_logistic_walk(.num_walks = 10) |>
  visualize_walks()

Student's t Distribution

random_t_walk()

Random walk with t-distribution (adjustable tail heaviness).

Function Signature:

random_t_walk(
  .num_walks = 25,
  .n = 100,
  .df = 5,
  .initial_value = 0,
  .ncp = 0,
  .samp = TRUE,
  .replace = TRUE,
  .sample_size = 0.8,
  .dimensions = 1
)

Parameters: - .df - Degrees of freedom (controls tail heaviness) - .ncp - Non-centrality parameter

Properties: - Heavier tails than normal (for low df) - Approaches normal as df → ∞ - Mean = 0 (if df > 1) - Variance = df/(df-2) (if df > 2)

Use Cases: - Robust statistics - Small sample inference - Financial returns modeling

Example:

# Heavy tails (df = 3)
random_t_walk(.num_walks = 10, .df = 3) |>
  visualize_walks()

# Nearly normal (df = 30)
random_t_walk(.num_walks = 10, .df = 30) |>
  visualize_walks()

Uniform Distribution

random_uniform_walk()

Random walk with uniformly distributed steps.

Function Signature:

random_uniform_walk(
  .num_walks = 25,
  .n = 100,
  .min = 0,
  .max = 1,
  .initial_value = 0,
  .samp = TRUE,
  .replace = TRUE,
  .sample_size = 0.8,
  .dimensions = 1
)

Parameters: - .min - Minimum value - .max - Maximum value

Properties: - All values equally likely - Mean = (min + max) / 2 - Variance = (max - min)² / 12

Use Cases: - Modeling equal probabilities - Random sampling - Monte Carlo simulations

Example:

# Symmetric around 0
random_uniform_walk(.num_walks = 10, .min = -1, .max = 1) |>
  visualize_walks()

Weibull Distribution

random_weibull_walk()

Random walk with Weibull distribution (flexible lifetime model).

Function Signature:

random_weibull_walk(
  .num_walks = 25,
  .n = 100,
  .shape = 1,
  .scale = 1,
  .initial_value = 0,
  .samp = TRUE,
  .replace = TRUE,
  .sample_size = 0.8,
  .dimensions = 1
)

Parameters: - .shape - Shape parameter (k) - .scale - Scale parameter (λ)

Properties: - Steps always positive - shape < 1: Decreasing failure rate - shape = 1: Constant failure rate (exponential) - shape > 1: Increasing failure rate

Use Cases: - Reliability engineering - Survival analysis - Wind speed modeling - Materials science

Example:

# Different shape parameters
random_weibull_walk(.num_walks = 10, .shape = 0.5) |>
  visualize_walks()

random_weibull_walk(.num_walks = 10, .shape = 2) |>
  visualize_walks()

Comparison Guide

When to Use Each Distribution

| Distribution | Use When... | Key Property | |--------------|-------------|--------------| | Normal | General modeling, natural phenomena | Symmetric, bell-shaped | | Brownian Motion | Continuous stochastic processes | Foundation of stochastic calculus | | Geometric Brownian | Stock prices, always positive | Log-normal, multiplicative | | Beta | Bounded processes (0-1) | Flexible shapes, bounded | | Cauchy | Extreme events, heavy tails | No mean/variance | | Chi-Squared | Sum of squares, variance tests | Right-skewed, positive | | Exponential | Time between events | Memoryless property | | F | Variance ratios | Ratio of chi-squared | | Gamma | Positive values, waiting times | Flexible, positive | | Log-Normal | Multiplicative processes | Right-skewed, positive | | Logistic | S-curves, classification | Heavier tails than normal | | Student's t | Robust modeling, small samples | Adjustable tail heaviness | | Uniform | Equal probabilities | All values equally likely | | Weibull | Reliability, survival | Flexible failure rates |

Tail Behavior

Light Tails (fewer extreme values): - Uniform

Medium Tails: - Normal - Exponential - Gamma (shape > 1)

Heavy Tails (more extreme values): - Logistic - Student's t (low df) - Cauchy (extreme)

Symmetry

Symmetric: - Normal - Brownian Motion - Cauchy - Logistic - Student's t - Uniform

Right-Skewed (tail extends right): - Beta (depends on parameters) - Chi-Squared - Exponential - F - Gamma - Log-Normal - Weibull (depends on shape)

Next Steps

• Discrete Distribution Generators - Explore discrete distributions
• Multi-Dimensional Walks - Work in 2D and 3D
• Visualization Guide - Create beautiful plots
• Use Cases and Examples - Real-world applications

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Need help choosing? Check out the FAQ or Use Cases for guidance!

RandomWalker documentation built on Sept. 8, 2026, 1:06 a.m.