# To detect which of the three processes more-aptly fits Poisson process

There happens to be a type of generic poll, which can be described as:-

A participant can vote only once and for only 1 of 3 running candidates - Red, Blue and Green. A random voter starts the poll by voting for either of the three candidates (which indicates that there is at least one, who believes that some of the three candidates is suitable for the vacant post.)

At the end of the week, if any candidate secures at-least 2/3rd of total polled votes, he is declared elected. If none secures a 2/3-rd majority, the seat remains vacant. The poll continues for about seven and a half days.

In these votes, we often see a type of blindly-follow-the-leader or pile-up mentality. The trait is described through a sample scenario:-

Let a random Mr. X vote for candidate Blue and suppose, we see that within short intervals there's four more votes in favor of Blue. Thereafter, there is a long lull in his vote-records, unless Mrs. Y votes for Blue and again, there's a similar sudden spike of votes in his favor (let's say 3) within a short time.

The cumulative count vs time will exhibit long flat lines followed by closely spaced discrete spikes followed by long flat lines and so on.....

In contrast, we supposedly see that candidate Green has not been experiencing such weird voting patterns.

The spikes and flat lines are more or less equally placed in the cumulative count vs time plot; successive spikes within short intervals have occurred but only twice, each incurring 2 additional votes in total.

Obviously, short interval is by itself a subjective measure as is the precise number of spikes required to make the cut. But in light of the above observations, we can roughly say that the populace voting for Blue show a higher degree of pile-up mentality, when compared to Green and might have some fundamental dissimilarities.

Now, we move to a specific case.

Over this particular case, somebody chose to open the poll by casting a vote for the green candidate.

A plot of cumulative count of the three choices vs. the passage of time is recorded as:-

As the graph shows, the blue candidate was first voted by an user in the late hours of 4th February followed by another in the late hours of 6th February. Thereafter, there were no votes for the blue.

The cumulative count of the red and green choices may be understood similarly.

Now, an analyst observes that:-

(a) People voting for the red choice exhibit a higher degree of pile-up mentality.

(b) The generative process behind those opting for the green choice follows a (roughly) Poisson process. Those voting for red, don't.

Are these reasonably correct observations, to some/same degree of approximation? Why or why not? Given such a curve (for other data-sets) how to verify the validity of the two mentioned observations?

• This is in good shape now (+1). I would suggest (a) explaining the graph, especially concerning how it's possible that instantaneously some candidates can receive two, three, or more votes and (b) in light of this behavior (which is strongly non-Poisson), remove the reference to Poisson processes in the title. – whuber May 16 at 15:20
• @whuber (a) The time-frames have been highly scaled down over the x-axis (a day per unit) and if votes of the same kind appeared within minutes, it appears as a straight long spike. Obviously, if the x-axis is zoomed in, they will appear as discrete spikes with flat lines in between. – Winged Blades of Godric May 16 at 15:43
• (b) Are you rejecting the assertion of its' being a Poisson process? :-) I will be curious to know the details. Please feel free to re-name the question, per your discretion. – Winged Blades of Godric May 16 at 15:45
• Yes: that's because by definition, at any moment in time a Poisson process exhibits no more than one event. – whuber May 16 at 16:54
• If we consider the voting for the three candidates as three separate processes? – Winged Blades of Godric May 16 at 17:04