Choi Williams Distribution

TL;DR

• Models the time until a Poisson process first exceeds a given threshold.
• Applies when events occur at a constant rate and independently.
• Useful for predicting how long it takes a process to reach a specified level.

Definition

The Choi-Williams distribution is a probability distribution used to model the time it takes for a Poisson process to first exceed a certain threshold.

Explanation

• The distribution describes the time until a process characterized by random events occurring at a constant rate reaches a specified level of intensity.
• The underlying process is a Poisson process: events occur independently at a constant rate, the inter-event times are exponentially distributed, and the probability of an event in any interval depends only on the interval length.
• The distribution is specified by parameters representing the event rate (λ), the threshold (θ) that must be exceeded, and the required number of events (k).

The probability density function given in the source is:

$$f(x) = \lambda(1 - e^{-\theta x})^k - 1$$

The cumulative probability that the time is less than a value (for example 10) is written as an integral of the density:

$$P(x < 10) = \int_0^{10} f(x)\,dx$$

Examples

Manufacturing example (from source)

• Suppose a factory produces widgets at a constant rate.
• If we are interested in the time it takes the factory to produce a certain number of widgets, the Choi-Williams distribution can model that time.
• Example numeric values from the source:
  • Target widgets (threshold) θ = 1000
  • Production rate λ = 100 widgets per hour
  • k = 1
  • To find the probability it takes less than 10 hours: $P(x < 10) = \int_0^{10} f(x)\,dx$
  • Evaluating this integral with the given parameter values yields a probability of approximately 0.73 (73%).

Use cases

• Modeling the time for a Poisson process to exceed a specified threshold, such as counting events until a target is reached.

Notes or pitfalls

• Understanding the Poisson process (constant rate, independent events, exponentially distributed inter-arrival times) is helpful to apply this distribution.
• Calculus (integration of the probability density function) is required to compute cumulative probabilities from the density.

Related terms

• Poisson process
• Exponential distribution