Draw best fitting Maxwell Boltzmann PDF

I got a dataset with values corresponding to velocities.

I need to obtain the PDF for those velocities. I'm not sure if I'm doing it correctly but here's my code for that:

input1 = dlmread('velocities_test.txt', ''); %dataset
input1([1,1],:) = [];
col1 = input1(:,1); %col1 has the values I want
[p,t] = hist(col1, 30); 
d = (p/sum(p)); %this is my PDF

After that I want to draw the best fitting Maxwell Boltzmann PDF for the PDF I obtained before.

The formula is the following:

What I need to do is to try several values for the 'a' parameter and find the best fitting ecuation.

a = (1.0:0.01:5.0);  % generating values for a between 1.0 and 5.0
for i=1:length(a)
  for j = 1:length(t)
      %Matriz de valor de 'ym'. Columnas = y(t) Filas = Cada Ym
     y(i,j) = sqrt(2/pi)*((t(j)*exp((-t(j)^2)/(2*(a(i)^2))))/(a(i)^3)) ; %this is the formula staten above
  endfor
endfor

figure(1)
for i = 1:length(a)
plot(t,y(i,:),'m-')
hold on  
end 
xlabel('Velocidad [m/s]')
ylabel('y(t)')
figure(1)
plot(t,d,'r-', "linewidth", 2)
set(gca, 'FontSize', 20)

Turns out this looks terrible and none of the functions seems to be the best fitting one.

Here they are in comparison to the original values:

What can I do to make the pink lines better? They look so small in comparison to the red one.

[the cyclist]

Not directly answering your question, but ...

Is there a reason you are doing this "fitting" manually, rather than using one of MATLAB's fitting functions (e.g. fitnlm from the Statistical and Machine Learning Toolbox).

Answer 1

[the cyclist]

It looks like the term ahead of the exponential should be

t(j).^2

rather than

t(j)

[Alejandro Santoflaminio]

Yes, that seems to be it! Thanks.