Understanding Poisson Properties for Client Arrivals in a Shop

[Mark J.]

Actually describing the process of arrivals for clients in a shop:
Getting difficulties with math describing of the process:

I wrote down three properties of Poisson but adviser is asking to describe very carefully how we understand these properties to reason that process is Poisson without making this assumption from the beginning.

Any help to get out from this abstraction??

Regards

 

[CompuChip]

So what are the three properties that you have?
I assume they are quite mathematical (something about independent events, blahblah, ...). Can you translate that into real-world language (involving customers and waiting times)?

 

[Mark J.]

Independence, memory less etc.
I am trying to do the translation bu always failing on it :)

 

[CompuChip]

OK, so let's try together.
Independence means that the occurrence of one event does not have a causal correlation to another event, right?

What is "the occurrence of an event" in your case?

 

[Mark J.]

arrival of one client:)

 

[CompuChip]

Good, so let's "plug that in":

Independence means that the arrival of one client does not have a causal correlation to the arrival of another client.

Does that make sense to you?

 

[Mark J.]

It does but for the adviser is not acceptable just to write they do not have a causal correlation because there is a "why" after that

 

[CompuChip]

That's a sensible question, isn't it?
I mean, if you were modelling people arriving at a school or office building throughout the day, I wouldn't say that events are independent.

 

[Mark J.]

Of course not but there is need for further words on this I think.

 

[CompuChip]

Well apparently your adviser thinks otherwise. So you should either make an effort to explain why this is different, or argue with him why it's too obvious. I'm just trying to guide you through it :)

 

[Mark J.]

Yes thank you please continue

 

[chiro]

Can you thinking of any kind of physical process that might cause some process dependencies between the two observations?

What kinds of attributes are common to the processes of multiple observations and how do they contribute to the process creating the observations?

 

[Mark J.]

No I cannot think of one process that connects passenger arrivals at the bus stop because schedule of bus is not fixed.Anyway its kind of strange just to say not correlated.Thought maybe can be a more specific math speech style

 

[chiro]

What kind of attributes do you attach to the process of a passenger arriving? (Not mathematical or quantitative at the moment, but qualitative and descriptive).

Also (and this is crucial with things like this): what kind of constraints and relationships do you put on the clients themselves? Are there properties of the clients that create some kind of bias and dependencies between arrivals?

 

[Mark J.]

Sorry I missed you here.
Maybe you mention an example?

 

[chiro]

One brief example that I can give is customer type.

Customers that go to a particular store may be frequent customers or they may be customers that visit at particular times of the day depending on what keeps them occupied.

For example 9-5 workers may get time to shop after work and not during the day where-as retired people, students, stay at home people, etc may go during the day.

What is sold will also affect the clients and arrivals.

The season will also affect things (Valentines Day with chocolates and Roses).

Also you must consider what is sold: something like a petrol or gas station will have very different properties and arrival times to even that of a super-market.

These are a few examples highlighting how things can cause dependencies between observations and other factors.

 

[Mark J.]

No I cannot think anything of these for passenger arrivals.
They are completely random in my case.
Only time of observation is peak hour in the morning.

 

[haruspex]

One brief example that I can give is customer type.

Customers that go to a particular store may be frequent customers or they may be customers that visit at particular times of the day depending on what keeps them occupied.

For example 9-5 workers may get time to shop after work and not during the day where-as retired people, students, stay at home people, etc may go during the day.

What is sold will also affect the clients and arrivals.

The season will also affect things (Valentines Day with chocolates and Roses).

Also you must consider what is sold: something like a petrol or gas station will have very different properties and arrival times to even that of a super-market.

These are a few examples highlighting how things can cause dependencies between observations and other factors.

Those are all examples of why the rate parameter may vary with time, but not of why one arrival may alter the probability (given the time of day) of another. Instead, consider that some people shop in pairs or even larger groups. Cluster arrivals would be distinctly non-Poisson.

 

[chiro]

What about the occurrence of retaining new customers?