Notes on the Poisson Distribution

Have you ever entered the grocery store in a rush to get just a few items and looked at the people in line at the fast cashier (12 items or less) and noticed there is no customers there? Then breathing a sigh of relief, rushed to get your few items? Then zoomed to the fast cashier only to see 6 customers in line?????

You have been hit in the eye, with the POISSON!

We live in an approximate Poisson world, not an approximate uniform world.

As an illustration consider the following example.

Suppose we work in an expensive camera store and have noticed that on the average 10 customers enter the store in one hour.

If we let X denote the number of customers that enter the store in one hour then X is Poisson with a mean of 10. This means that the average interarrival times between customers is 1/10 of an hour or 6 minutes. That is, about every 6 minutes we expect some customer to walk through the door.

Here are interarrival times in minutes of the first 20 customers (read row by row):

    1.7650    1.3700    3.1986   15.0311   16.1284    1.8612    1.0138
    1.1606   40.2126   15.1470    5.6882    5.3726    0.4688    1.4273
    6.9958    0.4205   10.9402    0.7203   17.8285    1.6083

So the first customer arrives in 1.7 minutes, the next 1.37 minutes later, etc. Here are the cumulative arrival times:

    1.765     3.135     6.334    21.365    37.493    39.354    40.368
    41.529    81.741    96.888   102.577   107.949   108.418   109.845
   116.841   117.262   128.202   128.922   146.751   148.359

Notice that by 41.529 minutes, 8 customers have arrived. Not bad, about what we expect, actually a bit better. For even two more customers should arrive in about 12 minutes and 41.529 plus 12 is still less than an hour. So we are having a great morning. But notice the next customer, Customer 9, takes 40.2126 minutes more to come. The store has been empty of customers for 40 minutes or so (depends on service time) and the help are standing around IDLE. The owner is losing money.

Now go back to the first customers. The first three should have entered in the first 18 minutes. In a uniform world they would be (on the average) equally spaced apart at 6 minute time intervals. They didn't confirm to the uniform model. Instead of 6 + 6 + 6 = 18 minutes, they arrived in 1.7 + 3.1 + 6.3 = 11.1 minutes. TOO MANY CUSTOMERS. Customers get tired of waiting for service and LEAVE. The owner is losing money.

People like to think of interarrival times as uniform. They aren't. They are Poisson.

The interarrival times are far from uniform. Here's a histogram of 100 of them:

Midpoint   Count
        0      45  *********************************************
        5      33  *********************************
       10      10  **********
       15       8  ********
       20       2  **
       25       0
       30       0
       35       0
       40       1  *
       45       1  *
And a boxplot:
            -------
           -I +   I------ ** * *                   O      O
            -------
           +---------+---------+---------+---------+---------+------C23
           0        10        20        30        40        50

 
In order to get a better idea of the lower interarrival times, I removed the two largest and we did the histogram:
 Midpoint   Count
        0      13  *************
        2      35  ***********************************
        4      16  ****************
        6      13  *************
        8       2  **
       10       4  ****
       12       5  *****
       14       2  **
       16       6  ******
       18       1  *
       20       1  *

These are still far from UNIFORM.

In general, the interarrival times of the world are not uniform.