# How can obtain the probability density function for a random discreate set of data and fit a custom distribution function ??

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Surendra Ratnu on 8 Dec 2023
Commented: Surendra Ratnu on 14 Dec 2023
I have random data. I want to calculate the probability density function. I also fit a custom distribution function which is the where gamma is the gamma function of n and n is the distribution parameter and x is the variable.
I attached the data. Can you provide the solution for this ??
Torsten on 8 Dec 2023
Edited: Torsten on 8 Dec 2023
Surendra Ratnu on 9 Dec 2023
Edited: Surendra Ratnu on 9 Dec 2023
@Paul It is true P(n,x) = 0 for x<0.

Torsten on 8 Dec 2023
Edited: Torsten on 8 Dec 2023
Use "histogram" to plot the empirical probability density function for your data and use "mle" to estimate n.
Why are there so many data repeated multiple times ? Did you create them artificially ?
Torsten on 13 Dec 2023
pdf = @(x,n)n^n/gamma(n) * x.^(n-1) .* exp(-n*x);
[phat,pci] = mle(x,'pdf',pdf,'Start',1)
phat = 3.3776
pci = 2×1
3.1120 3.6431
hold on
h = histogram(x);
h.Normalization = 'pdf';
y = linspace(0,max(x),100);
plot(y,pdf(y,phat))
hold off
Surendra Ratnu on 14 Dec 2023
Thank you for your inout. It's works. How can i know the goodness of the distribution fit if i fit using mle??

John D'Errico on 12 Dec 2023
Edited: John D'Errico on 12 Dec 2023
Can you write custom code to fit a PDF to data? Yes. It is not that truly difficult, if you know what you are doing. HOWEVER, SHOULD you write custom code to fit a PDF using maximum likelihood estimation? NO. It already exists in MATLAB, Use those codes. Writing your own code to do something already provided to you is a bad idea. If you don't have a clue what you are doing and don't have the skills in such things, then your result will be poor. Use code provided to you by professionals. Even there, you can have problems if you use the code incorrectly or provide it bad data that does not fit the distribution, but at least you give yourself a chance of success.
Now, does your data have a problem? YES. It is not at all random looking. In fact, what you have is not even a set of samples from a distribution, but apparently a cumulative histogram of sorts.
plot(x)
In there I see what appears to be stairsteps in your "data". So these are not samples from a continuous distribution. For example, the first 54 points in that vector are identical. It is not clear how you generated this data, which you claim to have measured.
So first, you need to explain what you have in that data set. How you created it.
Surendra Ratnu on 13 Dec 2023
Actually, Data are droplet size of the spray. I measured the drop size of spray using some software. The spray created identical drop size that why the data are in repeatative manner.
Paul on 13 Dec 2023
I think writing one's own code to solve a problem is great way to learn. The "professional" code can be used for comparison.

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