Random walk inputs
Use a one-dimensional biased walk with configurable steps, target, and Monte Carlo simulations.
Example data table
| Scenario | Steps | Start | Up step | Down step | Up probability | Target | Simulations |
|---|---|---|---|---|---|---|---|
| Biased upward walk | 100 | 0 | 1 | 1 | 0.52 | 10 | 3000 |
| Neutral walk | 200 | 0 | 1 | 1 | 0.50 | 15 | 5000 |
| Large downside jumps | 120 | 5 | 1.5 | 2.5 | 0.60 | -8 | 4000 |
Formula used
Let each step be a random variable \( Y \) where the walk moves up by \( a \) with probability \( p \), and down by \( b \) with probability \( 1-p \). The position after \( n \) steps is:
\( X_n = X_0 + \sum_{i=1}^{n} Y_i \)
Expected step: \( \mu = p \cdot a - (1-p) \cdot b \)
Step variance: \( \sigma^2 = p a^2 + (1-p)b^2 - \mu^2 \)
Expected final position: \( E[X_n] = X_0 + n\mu \)
Final variance: \( Var(X_n) = n\sigma^2 \)
Final standard deviation: \( \sqrt{Var(X_n)} \)
The calculator also runs Monte Carlo simulation to estimate hit probability, percentile ranges, median endpoint, and average first-passage timing. Endpoint target probability is approximated using a normal model.
How to use this calculator
- Enter the total number of steps in the walk.
- Set the starting position for the path.
- Enter the size of each upward and downward move.
- Provide the probability of an up-step between 0 and 1.
- Choose a target level you want the walk to reach or exceed.
- Set the number of simulations for Monte Carlo estimation.
- Optionally enter a seed to reproduce the same simulated path.
- Click the calculate button to generate metrics, charts, and the path table.
- Use the CSV button for structured data export or the PDF button for a report snapshot.
FAQs
1. What does this random walk calculator measure?
It estimates drift, variability, target-reaching probability, and likely endpoint ranges for a one-dimensional random walk. It combines theoretical formulas with simulation-based evidence.
2. What is a biased random walk?
A biased random walk favors one direction more than the other. That bias may come from unequal probabilities, unequal step sizes, or both, which changes the expected endpoint.
3. Why does the calculator use simulations?
Simulation captures path-dependent behavior such as hitting a level during the walk, percentile ranges, and first-passage timing. Those details are harder to express with simple closed-form formulas.
4. What is hit probability?
Hit probability is the fraction of simulated paths that reach the target at any step before the walk ends. It differs from merely finishing beyond the target.
5. What does the endpoint probability represent?
It estimates the chance that the final position is beyond the target threshold after all steps are taken. This calculator reports both theoretical approximation and simulated endpoint success.
6. Why can the theoretical and simulated values differ?
Theoretical results describe idealized expectations, while simulations use random samples. Differences shrink as the number of simulations increases, though approximation limits can still remain.
7. What does the random seed do?
A seed fixes the pseudo-random sequence. Using the same seed and inputs reproduces the same simulated path and summary results, which helps testing and teaching.
8. Where can this calculator be used?
It is useful in probability lessons, finance intuition, queueing examples, algorithm analysis, diffusion-style models, and any scenario where repeated uncertain steps accumulate over time.