Random Walk & Brownian Motion Simulator
Simulate and visualize 1D random walks and Brownian motion paths with empirical distributions. Explore probability and stochastic processes through interactive plots.
Formula (1D Random Walk): \( S_n = \sum_{i=1}^n X_i, \; X_i \in \{-1, 1\} \)
Brownian Motion: \( B(t) \sim \mathcal{N}(0, t) \)
Simulation Results
Statistics
About Random Walks and Brownian Motion
A random walk is a mathematical object that describes a path consisting of successive random steps. The 1D random walk shown here takes steps of ±1 with equal probability.
Brownian motion is the continuous-time analog of the random walk, where the position changes according to a normal distribution with variance proportional to time.
These concepts are fundamental in probability theory, physics, finance, and many other fields.
Related Calculator Tools: 1) 3D Box Volume Calculator | Free Online Tool, 2) 3D Function Plotter | Math Visualization Tool 3) Confidence Interval Calculator | Accurate Stats Tool Online 4) Bayesian Calculator – Visualize Beta-Binomial Distributions, 5) Interactive Convolution Calculator – Visualize Signal Processing, 6) Fourier Transform Calculator | Signal Analysis Tool Online 7) Random Walk & Brownian Motion Simulator – Visualize Paths
How to Use Random Walk & Brownian Motion Simulator
Mastering the Random Walk Simulator in 3 Steps:
- Configure Your Simulation:
- Select either Random Walk or Brownian Motion mode
- Set steps (100-10,000) and paths (1-50)
- Adjust step size for random walks (default 1 unit)
- Run & Analyze:
- Click ‘Simulate’ to generate interactive paths
- Hover over graphs to examine specific points
- View automatically calculated statistics (mean, variance)
- Export & Share:
- Download visuals as SVG/JPG for presentations
- Screenshot statistical comparisons
- Bookmark custom parameter settings
Pro Tip: Compare multiple simulations side-by-side in different browser tabs to study probability distributions.
FAQs:
Q: What’s the difference between random walk and Brownian motion?
A: While both model random movement, random walks use discrete steps (±1) while Brownian motion shows continuous normal-distributed movement – our tool visualizes both!
Q: How accurate are the simulations?
A: Our algorithms use rigorous mathematical models – random walks employ Bernoulli trials while Brownian motion uses Box-Muller transforms for normal distribution.
Q: Can I use this for financial modeling?
A: Absolutely! Many traders use such simulations to model stock movements. For best results, run 50+ paths with 1,000+ steps.
Q: Why do my paths look different each time?
A: This demonstrates true randomness! Each simulation generates independent paths – try increasing path count to see convergence patterns.
Q: Is there mobile support?
A: Yes! Our responsive design works on all devices. For best experience, use landscape mode on phones when analyzing graphs.
Q: Are results exportable for research?
A: Certainly! Download SVG/JPG visuals or manually record statistics. For raw data, use browser developer tools.
Q: What’s the maximum simulation size?
A: We support up to 10,000 steps and 50 simultaneous paths – enough for most educational/research needs while maintaining performance.