My Data has negative and positive numbers, can I still use it to get the PDF?

Question

I have data that reflects the arrival time of a flight. {3, -2, 16, 13, -6, 5, 4, -7, 7, 0……..n} this is for 8:45am and has 150 elements "arrivals" The positive values represent the time in minutes past the planned time,{“late”} and the negative, is before planned time {“early”} My question, I would like to find the time of arrival with the highest probability. Can I use this data as is?

Answer 1

For the sake of giving a (hopefully) useful answer.

Lets generate 150 arrival times where negative values indicate early and positive times indicate late arrivals. Here I'm assuming that people are as likely to be late as early and the distribution of arrival times is BinomialDistribution[20, 1/2]. This further assumes that people tend to be about 10 minutes early or 10 minutes late.

SeedRandom[1];
arrival = 
 RandomChoice[{-1, 1}, 150]*
  RandomVariate[BinomialDistribution[20, 1/2], 150];

Now there are a number of things available to us in Mathematica for working with this data.

Lets compute the expected arrival time.

Expectation[a, a \[Distributed] arrival]

(* -(109/75) *)

Or we could say, what is the probability someone is late?

Probability[a > 0, a \[Distributed] arrival]

(* 13/30 *)

We can estimate distributions from the data. Here a mixture of normals is used which is a little silly given that the data is discrete but normal is known to be a reasonable approximation to the binomial distribution with large enough samples and we expect a bimodal distribution given the setup.

 est = EstimatedDistribution[arrival, 
 MixtureDistribution[{1/2, 1/2}, {NormalDistribution[a, b], 
   NormalDistribution[c, d]}]]

(* MixtureDistribution[{1/2, 1/2}, {NormalDistribution[9.75385, 2.15571], 
  NormalDistribution[-10.0235, 2.35635]}] *)

We can plot all sorts of things and assess the goodness of fit to our chosen distribution.

Show[Plot[PDF[est, x], {x, -25, 25}], SmoothHistogram[arrival]]

We should keep in mind here that the p-value will probably be larger than it should be since I've estimated the distribution from the data first...

DistributionFitTest[arrival, est]

(* 0.107467 *)

We can always check further with a QuantilePlotwhich claims it doesn't fit so well in the body of the distribution...

QuantilePlot[arrival, est]

The bottom line is that there is nothing special about negative arrival times as far as Mathematica is concerned. It is happy to work with them just like any other data.