L22.5 The Mean and Variance of the Number of Arrivals | Summary and Q&A

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April 24, 2018
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L22.5 The Mean and Variance of the Number of Arrivals

TL;DR

The mean number of arrivals in a Poisson process is equal to lambda times tau, while the variance is also lambda times tau.

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Q: What is the formula for the expected value of the number of arrivals in a Poisson process?

The expected value of the number of arrivals in a Poisson process is lambda times tau, where lambda is the arrival rate and tau is the length of the time interval.

Q: How does the binomial approximation help in calculating the expected value of the number of arrivals?

The Poisson process can be approximated by a binomial random variable with n and p parameters, and the expected value of the binomial random variable is n times p. In this case, n times p evaluates to lambda times tau.

Q: What is the formula for the variance of the number of arrivals in a Poisson process?

The variance of the number of arrivals in a Poisson process is also lambda times tau, where lambda is the arrival rate and tau is the length of the time interval.

Q: Why is lambda referred to as the arrival rate or the intensity of the arrival process?

Lambda represents the expected number of arrivals per unit time in a Poisson process. Since it is the expected number of arrivals divided by the length of time, it is a natural interpretation of the arrival rate or intensity.

Summary & Key Takeaways

• The expected value of the number of arrivals in a Poisson process is lambda times tau, based on a discretization argument and the approximation by a binomial random variable.

• The variance of the number of arrivals in a Poisson process is also lambda times tau, with a small impact from the approximation of the binomial distribution.