# Finding the optimal probabilty distribution

I am trying to obtain the optimal probability distribution function to the following data.

My attempt:

data = Import["data_prob_new.dat", "Table"];
P00 = Histogram[Flatten[data], Automatic, "Probability",
ChartStyle -> Gray, ChartBaseStyle -> EdgeForm[None],
ImageSize -> 500]
fit = FindDistributionParameters[Flatten[data],
LaplaceDistribution[a, b]];
a0 = a /. fit[[1]];
b0 = b /. fit[[2]];
lim = 1000;
t0 = Plot[PDF[LaplaceDistribution[a0, b0], x], {x, a0, lim},
PlotStyle -> {Blue, Thick}, PlotRange -> All];
P0 = Show[{P00, t0}, Frame -> True, Axes -> False,
FrameStyle -> Thick, PlotRange -> {{0, All}, {-0.001, All}},
PlotRangePadding -> 0, PlotRangeClipping -> True]


As we can see, the Laplace probability distribution fails to smoothly fit the tail of the histogram. My question: How can we obtain the best fit (type of distribution) for this histogram?

• The data are not distributed Laplace.
– Alan
Nov 3 '18 at 12:07
• Gamma distribution might be better Nov 3 '18 at 12:46
• Try using the EstimatedDistribution function, with a range of PDFs. Looks a bit like the exponential distribution to me. Nov 3 '18 at 15:21
• There is no "optimal" because you haven't stated what kind of process generated the data. You just have data for which you'd like a reasonable and more compact description such as "Exponential distribution with parameter $\lambda$". Also, you want "PDF" rather than "Probability" to make the histogram and probability density match in scale.
– JimB
Nov 3 '18 at 16:51
• "no good" ???? See answer below.
– JimB
Nov 3 '18 at 17:49

Given that you just have data and an urge to fit a parametric probability distribution, the quality of the fit is in the eye of the beholder. Here's the fit with the gamma distribution (suggested by @mikado) (which I think you'll be hard-pressed to find a better fit):

P00 = Histogram[data, Automatic, "PDF", ChartStyle -> Gray,
ChartBaseStyle -> EdgeForm[None], ImageSize -> 500,
PlotRange -> All];
fit = FindDistributionParameters[data,

• Just a comment: Inside the Histogram[...] you changed the option from "Probability" to "PDF". Why? What if we want to plot probability instead of PDF and find the best fit of it? Nov 3 '18 at 21:24
• You were estimating the parameters of a probability distribution function (pdf) and using the PDF function to display the probability distribution function. For a histogram to match scales, you need the "PDF" option. About the second question I answer with a question: Why would you want a probability scale that's associated with an arbitrary sample size and arbitrary layout of the bins?