Inverse Gaussian First Passage

Worked example: Finding the density

Let's calculate the density function of the inverse Gaussian distribution at x = 1, given parameters μ = 2 and λ = 1.

1. Identify the parameters: μ = 2, λ = 1, x = 1.
2. Calculate the first term: sqrt(λ / (2 * π * x³)) = sqrt(1 / (2 * π * 1)) ≈ 0.398942.
3. Calculate the exponent: -λ(x - μ)² / (2μ²x) = -1 * (1 - 2)² / (2 * 2² * 1) = -1 / 8 = -0.125.
4. Calculate the second term: exp(-0.125) ≈ 0.882496.
5. Multiply the terms: 0.398942 * 0.882496 ≈ 0.352065.

The independently calculated density is approximately 0.352065, matching the expected output.

Experiment: Changing the shape

You can observe how the distribution shape changes by adjusting the parameters in the Inverse Gaussian First Passage module.

1. Set the Mean mu to 1.
2. Change the Shape lambda from 1 to 4.
3. Observe that the variance decreases from 1 to 0.25, and the distribution becomes more concentrated around the mode.
4. Set the Probe x to 1 and verify the density increases from 0.3989 to 0.7978.

Limitations and Assumptions

The inverse Gaussian distribution assumes a constant positive drift and a constant diffusion coefficient for the underlying Brownian motion. If the drift is zero or negative, the process may never reach the target level, leading to a defective distribution where the total probability mass is less than 1. Additionally, this distribution is only defined for x > 0, as the first passage time must be strictly positive.

Primary References

The density and parameterization used in this module follow the standard statistical definition, commonly found in software libraries.

• SciPy Documentation: scipy.stats.invgauss. This page provides the mathematical formulation of the inverse Gaussian distribution, though SciPy uses a different parameterization internally.