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Conditional Jump Times of a Poisson Process Are Uniform

Article Quant Q&A · Author: Lost1
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Summary

The document concerns a probability result used in modeling event arrivals: conditional on a Poisson process having a fixed number of jumps within an interval, the jump times are uniformly distributed over that interval. The question asks for an accessible online proof rather than supplying a derivation or describing a trading application.

The answer points to a reference and identifies a page that may contain the result. It provides no proof details, assumptions, worked example, or discussion of how the fact might be used in finance. The document therefore serves mainly as a pointer to a foundational probability concept. Readers should consult the referenced mathematical treatment to confirm the precise conditioning statement and its setup before applying it to simulations or event-time models.

Key ideas

  • For a Poisson process, jump times conditional on a fixed count within an interval have a uniform distribution over that interval.
  • The document directs readers to an external mathematical reference for a proof.
  • No derivation, example, or financial application is included in the exchange.
  • The result can inform probabilistic modeling of event arrival times, subject to the assumptions of the Poisson process.

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# Prove the jump times of a poisson process in a given interval are uniformly distributed


# Prove the jump times of a poisson process in a given interval are uniformly distributed












Can someone provide a reference for this fact on the internet?

While I know this fact is proved in many text books, but I cannot find a proof of this fact very easily on the internet

## Answer by Mehness (score 3, accepted)

https://quant.stackexchange.com/a/31282

http://www.math.tau.ac.il/~uriy/Papers/encyc57.pdf - page 5 in this does that do it?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.