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Essentials of Mathematical Probability and Statistics

John Travis

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Section 8.1 Time vs Changes

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With Bernoulli-based distributions, you considered the number of successes in a fixed number of trials (Binomial) or the number of trials until obtaining a fixed number of successes (Negative Binomial/Geometric). For each of these, everything was discrete--trials and successes. Another option is to investigate the relationship between number of successes (or "changes") relative to some interval of time. That is, the number of trials for Bernoulli-based distributions is replace with an interval of time. This approach involves the discrete number of changes with a continuous interval of time.
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Definition 8.1.1. Poisson Process.

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Consider an experiment that takes place over some interval of time. A Poisson process is a course of action in which:
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    Successes in non-overlapping subintervals are independent of each other.
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    The probability of exactly one success in a sufficiently small interval of length h is proportional to h. In notation, P(one success) = Ξ»h.
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    The probability of two or more successes in a sufficiently small interval is essentially 0.
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These assumptions lead to the following distributions discussed in this chapter.
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In this chapter, you will investigate the following distributions:
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    Poisson - the interval is fixed and X measures the variable number of successes.
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    Exponential - the number of successes is fixed--at 1--and X measures the variable interval length needed to get that success.
  3. πŸ”—
    Gamma - the number of successes is fixed and X measures the variable interval needed to get the desired number of successes.
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