Poisson Distribution Question

#1
I am using the Poisson distribution model the determine the probability of an event occurring. I know that the mean of the event occurring is 7. When I look at the Poisson distribution table, it says that there is a 45.0% chance of the event happening less than 7 times, 14.9% chance of the event happening exactly 7 times, and a 40.1% chance of the event occurring more than 7 times.

To me, it does not make logical sense that that there is a bias towards the under 7. Why would the distribution not be evenly spread above and below 7? What logical sense does it make for something with a mean of 7 having a larger chance of happening less than 7 times than more than 7 times?

Thanks in advance for any help.
 
#2
Hi Relentless,

Welcome to the forum.

The poisson distribution resembles the binomial distribution if the probability of an event is very small. Generaly, for small values of mean, the distribution is not symmetric but skewed. The distribution becomes more symmetric when the mean is larger.

You can try Poisson with mean=1,3,5,7... and see the trend.
 
#3
Thanks Quark.

Why is the distribution skewed though - is it skewed because of a flaw in the Poisson distribution or does this reflect accurate real-world expectations?
 
#5
If possible could you help me understand the logic of this. Using the example above, Why if something has an average of 7 would it be more likely to occur less than 7 times than more than 7 times? Why wouldnt it be logical to assume that it is equally likely for it to occur more/less than the mean?
 
#6
If you plot P(x) vs x you can see that the distribution is asymmetric.

If you have a Poisson with mean of 1, You would have:

x P(x)
0 0.367879
1 0.367879
2 0.183940
3 0.061313
4 0.015328

The distribution is not symmetric about 1. You wouldn't expect the probability to be the same for x=0 and x=2. Poisson random variables are non-negative. You still have nonzero probabilities for x=3,4,...

When the mean=7, it's less extreme but the same logic holds.
 

JohnM

TS Contributor
#7
Another way to look at this -

if we are modeling the number of times an event occurs during a given time period, it is impossible for this event to occur fewer than 0 times, so the distribution cannot go below 0.

On the other side, theoretically the event could occur an infinite number of times, so it is "unbounded" to the right.

So, this expains the shape of the distribution, in a sense - it can't go below 0, but it can get infinitely large.
 
#8
I really appreciate the help guys. I am not questioning that this mathematical fact, but I just can't wrap my head around the real-world logic of it.

If you sit by the road and count the cars that go by over a 1 hour time span and determine the mean to be 100. Poisson says that over the next hour if you watch again there is a greater chance (48.7 v 47.3) of there being less cars than 100. I just can't understand this - I would think there would be a an even chance of there being more or less cars. Should I not be using Poisson for this type of problem?

Sorry, I seem to be asking the same question over and over, but I just do not understand the real-world logic of the Poisson distribution. Is binomial distribution a better tool to use for examples such as these or is there little difference between the two types of distributions?
 

JohnM

TS Contributor
#9
The mean of a distribution doesn't necesarily equate to the "middle" of a distribution - it's actually the "center of gravity."

With your car/traffic example, if the mean were 100, I can understand why it is difficult to visualize why there is >50% chance of the number of cars being less than 100.

It's probably easier to see when modeling events that have means much lower than this.
 
#10
Hi Relentless,

I think you are thinking along the lines of normal distribution, which is symmetric. Poisson is not symmetric, and the number of counts in a set period of time is usually distributed as Poisson. Just like John said, you won't see negative number of cars, but you may see a HUGE number of cars (say 1000), although the probability is small.