I thought a Poisson distribution was an exponential distribution that began at 0, rose sharply to a peak, and then had a very long tail .
I wanted to see the numbers produced by the RANPOI function, so I did a simple bit of code in EG
do i=1 to 10000;
metric = ranpoi(0,50);
and then simply plotted a bar chart of metric.
What I got looks a lot more like a normal distribution than what I thought I would get.
People, please set me straight on what is going on here, please.
The shape of the Poisson distribution changes with the value of the mean. With a relatively small value for the mean like a 1, the distribution has an exponential shape that you mentioned. But when the mean gets large it starts to approximate the binomial distribution. I forgot the details, my stats classes were many years ago, so this example below should shed some light on the matter.
do i=1 to 10000;
metric1 = ranpoi(0,1);
metric5 = ranpoi(0,5);
metric10 = ranpoi(0,10);
metric50 = ranpoi(0,50);
I did some refresher research.
I see where I was making a mistake in my interpretation.
Now what I find to be aggrevating with the SAS's implementation of Poisson, is that it has no provision for lambda.
So, here is where I need help.
I want to model some aspects of human behavior. The variable is inter-arrival times. Most intervals between events is about 10ish seconds (9 to 12). Some intervals can be as long at 5 minutes (long right tail). This is for a single even/impulse stream. I want the intervals between impulses/events to be randomly generated per an appropriate distribution. I then want to vary the number of concurrent impulse streams.
How do I do this?
Shouldn't I be able to use a Poisson distribution? Or am I mis-interpreting something?
In the past, I just used real data and "replayed" it to do whatever analysis I needed to do. This time, I don't have that ability/luxury, and I need to actually synthetically and arbitrarily create the impulse/event streams.
I think you are mis-interpreting the use of a Poisson distribution, but I'm no expert and I haven't done any time-domain analysis in while. So take what I say with a grain of salt.
The SAS ranpoi function has two paramters ranpoi(seed, m) where seed is some arbitray seed value and m is the mean # of events. With the Possion distribution, lambda is the mean # of events, so m=lambda for the SAS function.
Notice that the results of using ranpoi is an integer, i.e. # of events. This is a crucial distinction when interpreting the Poission distribtion. It measures the # of events within a fix time period, not the time between events. So I don't think the Possion distribution will not work for you. Unless of course you reframe your question to the # of events/impulses per second or minute.
Now for the rub, what you discribe is a Possion process but you should be using an exponential distribution for your analysis/simulation. Not all Possion processes can be modeled with a Possion distribution.