Model first arrivals across random walks, diffusion, counts. Test scenarios with examples and clean exports. Get reliable estimates using clear inputs and simple steps.
| Model | Inputs | Expected Hitting Time | Extra Output |
|---|---|---|---|
| Gambler ruin | N = 10, i = 4, p = 0.50 | 24.000000 steps | Upper boundary probability = 0.400000 |
| Boundary walk | L = 0, U = 12, x = 5 | 35.000000 steps | Fair walk to either boundary |
| Poisson threshold | lambda = 2.5, k = 8 | 3.200000 time units | Variance = 1.280000 |
| Brownian drift | a = 6, mu = 1.5, sigma = 2 | 4.000000 time units | Standard deviation = 2.666667 |
For a fair walk, use E[T] = i(N - i). For a biased walk, use E[T] = i/(q-p) - [N/(q-p)] * [(1-(q/p)^i)/(1-(q/p)^N)], where q = 1 - p.
For absorbing boundaries L and U, the expected hitting time from x is E[T] = (x - L)(U - x).
The first time to reach k events has mean E[T] = k / lambda and variance Var(T) = k / lambda^2.
For X(t) = mu t + sigma W(t), the mean time to hit level a is E[T] = a / mu when mu is positive. The variance is a sigma^2 / mu^3.
Hitting time is the first moment a random process reaches a target state. Many texts also call it first passage time. The idea appears in probability, finance, queue theory, physics, biology, and reliability analysis. It answers a simple question. How long until a threshold is crossed?
A hitting time calculator helps turn theory into fast estimates. Students use it to check homework and intuition. Analysts use it to test thresholds, waiting times, and absorption events. Researchers use it when they compare random walk models, diffusion models, and count processes. The result gives a practical summary of when a state is likely to be reached.
This page includes four common models. The gambler ruin section handles absorption between two boundaries. The symmetric boundary walk section estimates the expected steps to hit either side. The Poisson model studies the time needed to collect a fixed number of events. The Brownian motion model estimates first arrival to a level when drift pushes the process forward.
The expected hitting time is an average value. It is not a guaranteed finish time. Real sample paths can arrive earlier or later. That is why variance and standard deviation matter in continuous and count models. A larger spread means more uncertainty around the average. Boundary probabilities also matter in biased walks because direction changes the outcome.
Hitting time appears in barrier monitoring, ruin analysis, queue buildup, neuron firing, event arrivals, and machine learning threshold checks. It also helps with diffusion based systems and threshold detection in noisy data. A clear calculator saves time and reduces manual errors. Use the formulas, example table, and export tools to document each scenario cleanly.
It is the first time a stochastic process reaches a chosen state, level, or boundary. It is also called first passage time in many probability books.
No. Some models can have infinite expected mean time under certain settings. In this calculator, Brownian motion with nonpositive drift to a positive level is one example.
N sets the full state range. i gives the starting position inside that range. Together they define how far the walk sits from both absorbing boundaries.
It is the probability that the walk hits the upper boundary before the lower boundary. Bias in the step probability changes this value.
Use it when events arrive randomly at a constant average rate and you want the expected waiting time until a target number of events appears.
Sigma controls path variability. A larger sigma means wider fluctuations. It changes the spread of the hitting time, even when the mean stays tied to the drift.
They are closed-form results for the listed model assumptions. The answer is exact for those assumptions, but real systems may require more detailed modeling.
Yes. After calculation, use the CSV button for data records or the PDF button for a simple printable summary of inputs, outputs, and formula details.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.