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