The Poisson Distribution

4. The Poisson Distribution

The Density Function

We have shown that the k'th arrival time has the gamma density function with shape parameter k and rate parameter r:

f_k(t) = (rt)^{k - 1}re^-rt / (k - 1)!, t > 0.

Recall also that at least k arrivals come in the interval (0, t] if and only if the k'th arrival occurs by time t:

N_t k if and only if T_k t.

$Mathematical Exercise$ 1. Use integration by parts to show that

P(N_t k) = _(0,
t] f_k(s)ds = 1 -_{j
= 0, ..., k - 1} exp(-rt) (rt)^j / j!.

$Mathematical Exercise$ 2. Use the result of Exercise 1 to show that the density function of the number of arrivals in the interval (0, t] is

P(N_t = k) = e^-rt (rt)^k / k! for k = 0, 1, ...

The corresponding distribution is called the Poisson distribution with parameter rt; the distribution is named after Simeon Poisson.

3. In the Poisson experiment, vary r and t with the scroll bars and note the shape of the density function. Now with r = 2 and t = 3, run the experiment 1000 times with an update frequency of 10 and watch the apparent convergence of the relative frequency function to the density function.

The Poisson distribution is one of the most important in probability. In general, a discrete random variable N in an experiment is said to have the Poisson distribution with parameter c > 0 if it has density function.

g(k) = P(N = k) = e^-c c^k / k! for k = 0, 1, ...

$Mathematical Exercise$ 4. Show that g really is a density function.

$Mathematical Exercise$ 5. Show that P(N = n - 1) < P(N = n) if and only if n < c

Thus, the distribution is unimodal, with the mode occurs at the greatest integer in c.

$Mathematical Exercise$ 6. Suppose that requests to a web server follow the Poisson model with rate r = 5 per minute. Find the probability that there will be at least 8 requests in a 2 minute period.

$Mathematical Exercise$ 7. Defects in a certain type of wire follow the Poisson model with rate 1.5 per meter. Find the probability that there will be no more than 4 defects in a 2 meter piece of the wire.

Moments

Suppose that N has the Poisson distribution with parameter c. The following exercises give the mean, variance, and probability generating function of N.

$Mathematical Exercise$ 8. Show that E(N) = c

$Mathematical Exercise$ 9. Show that var(N) = c

$Mathematical Exercise$ 10. Show that E(u^N) = exp[c(u - 1)] for s R.

Returning to the Poisson process, it follows from that

E(N_t) = rt, var(N_t) = rt.

Once again, we see that r can be interpreted as the average arrival rate. In an interval of length t, we expect about rt arrivals.

11. In the Poisson experiment, vary r and t with the scroll bars and note the location and size of the mean/standard deviation bar. Now with r = 3 and t = 4, run the experiment 1000 times with an update frequency of 10 and watch the apparent convergence of the sample mean and standard deviation to the distribution mean and standard deviation, respectively.

$Mathematical Exercise$ 12. Suppose that customers arrive at a service station according to the Poisson model, at a rate of r = 4 per hour. Find the mean and standard deviation of the number of customers in an 8 hour period.

Stationary, Independent Increments

Let us see what the basic regenerative assumption of the Poisson process means in terms of the counting variables.

$Mathematical Exercise$ 13. Show that if s < t, then N_t - N_s = the number of arrivals in (s, t]

Recall that our basic assumption is that the process essentially starts over at time s and the behavior after time s is independent of the behavior before time s.

$Mathematical Exercise$ 14. Argue that:

N_t - N_s has the same distribution as N_t-s, namely Poisson with parameter r(t - s).
N_t - N_s and N_s are independent.

$Mathematical Exercise$ 15. Suppose that N and M are independent Poisson variables with parameters c and d respectively. Show that N + M has the Poisson distribution with parameter c + d.

Give a probabilistic proof, based on the Poisson process.
Give a proof using density functions.
Give a proof using moment generating functions.

16. In the Poisson experiment, select r = 1 and t = 3. Run the experiment 1000 times, updating after each run. By computing the appropriate relative frequency functions, investigate empirically the independence of the random variables N₁ and N₃ - N₁.

Normal Approximation

Now note that for k = 1, 2, ...

N_k = N₁ + (N₂ - N₁) + ЗЗЗ + (N_k - N_k-1).

The random variables in the sum on the right are independent and each has the Poisson distribution with parameter r.

$Mathematical Exercise$ 17. Use the central limit theorem to show that the distribution of the standardized variable below converges to the standard normal distribution as k increases to infinity.

(N_k - kr) / (kr)^1/2.

A bit more generally, the same result is true with the integer k replaced by the positive real number t.

18. In the Poisson experiment, set r = 1 and t =1. Increase r and t and note how the graph of the density function becomes more bell shaped.

19. In the Poisson experiment, set r = 5 and t = 4. Run the experiment 1000 times with an update frequency of 100. Compute and compare the following:

P(15 N₄ 22).
The relative frequency of the event {15 N₄ 22}.
The normal approximation to P(15 N₄ 22).

$Mathematical Exercise$ 20. Suppose that requests to a web server follow the Poisson model with rate r = 5 per minute. Compute the normal approximation to the probability that there will be at least 280 requests in a 1 hour period.

Conditional Distributions

$Mathematical Exercise$ 21. Let t > 0. Show that the conditional distribution of T₁ given N_t = 1 is uniform on (0, t). Interpret the result.

$Mathematical Exercise$ 22. More generally, given N_t = n, show that the conditional distribution of T₁, ..., T_n is the same as the distribution of the order statistics of a random sample of size n from the uniform distribution on (0, t).

Note that the conditional distribution in the last exercise is independent of the rate r. This result means that, in a sense, the Poisson model gives the most "random" distribution of points in time.

$Mathematical Exercise$ 23. Suppose that requests to a web server follow the Poisson model, and that 1 request comes in a five minute period. Find the probability that the request came during the first 3 minutes of the period.

24. In the Poisson experiment, set r = 1 and t = 2. Run the experiment 1000 times, updating after each run. Compute the appropriate relative frequency functions and investigate empirically the theoretical result in Exercise 5.

$Mathematical Exercise$ 25. Let 0 < s < t and let n be a positive integer. Show that the conditional distribution of N_s given N_t = n is binomial with parameters n and p = s/t. Note that the conditional distribution is independent of the rate r. Interpret the result.

$Mathematical Exercise$ 26. Suppose that requests to a web server follow the Poisson model, and that 10 requests come during a 5 minute period. Find the probability that at least 4 requests came during the first 3 minutes of the period.

Estimating the Rate

In many practical situations, the rate r of the process in unknown and must be estimated based on observing the number of arrivals in an interval.

$Mathematical Exercise$ 27. Show that E(N_t / t) = r and hence N_t / t is an unbiased estimator of r.

Since the estimator is unbiased, the variance measures the mean square error of the estimator.

$Mathematical Exercise$ 28. Show that var(N_t / t) = r / t and hence var(N_t / t) decreases to 0 as t increases to infinity.

29. In the Poisson experiment, set r = 3 and t = 5. Run the experiment 100 times, updating after each run.

For each run, compute the estimate of r based on N_t.
Over the 100 runs, compute the average of the squares of the errors.
Compare the result in (b) with the variance in Exercise 26.

$Mathematical Exercise$ 30. Suppose that requests to a web server follow the Poisson model with unknown rate r per minute. In a one hour period, the server receives 342 requests. Estimate r.