Application: Poisson Processes

• Events occur continuously over time according to a Poisson process with rate \(\lambda>0\) (per unit time) if
  • The number of events that occur in any time interval of length \(t\) has a Poisson\((\lambda t)\) distribution.
    • In particular, the distribution does not depend on the starting time or ending time of the interval, just its length. This is called the stationary increments property.
  • The numbers of events that occur in disjoint intervals of time are independent.
    • This is called the independent increments property.
• A stochastic process is a Poisson process with rate \(\lambda\) if and only if
  • it is a counting process, that is, a discrete state stochastic process which counts the number of occurrences of some event over time.
  • the interarrival times (times between events) \(W_0, W_1,\ldots\) are independent Exponential\((\lambda)\) random variables

Arrivals of spam emails to your email spam filter follow a Poisson process with mean rate 1.5 spam emails per minute. For the parts below, in addition to computing/deriving, denote the corresponding objects in terms of proper symbols and notation.

In addition to answering the questions below, complete the companion Colab notebook.

1. Is a Poisson process discrete or continuous time? Discrete or continuous state? Explain briefly.

2. Do the interarrival times themselves comprise a discrete or continuous time process? Discrete or continuous state? Explain briefly.

3. Explain how you could use an Exponential(1) spinner to simulate a single sample path of the process.

4. Compute the probability that exactly 4 spam emails arrive to the filter in the next 2 minutes.

5. Compute the conditional probability that more than 3 minutes elapse, starting now, before the next spam email arrives.

6. Compute the conditional probability that more than 3 minutes elapse, starting now, before the next spam email arrives, given that the most recent spam email arrived 2 minutes ago.

7. Are the random variables representing the process values at times 0.5 and 0.6 independent? Explain briefly.

8. Are the random variables representing the arrival times of the first and second spam emails independent? Explain briefly.

9. Derive the mean function of the process.

10. Derive the variance function of the process.

11. Is the process wide sense stationary (WSS)? Explain briefly.

    You can do the next four parts in whichever order you prefer, just be sure to avoid circular reasoning. Hint: for \(t>s\), write \(N(t) = N(t) - N(s) + N(s)\) and take advantage of the independent increments property.

12. Derive the value of the autocorrelation function at times 0.5 and 0.6.

13. Derive the autocorrelation function of the process.

14. Derive the value of the autocovariance function at times 0.5 and 0.6.

15. Derive the autocovariance function of the process.

16. Find the value of the correlation coefficient at times 0.5 and 0.6.

17. Find the correlation coefficient between the process values at times \(t\) and \(s\), with \(s<t\). For fixed \(s\), how does the correlation coefficient behave as \(t\) increases? Does this make sense?

18. Suppose that exactly 1 spam email arrives in the next 2 minutes. Derive the conditional probability that this email arrives in the next 30 seconds.