[SOLVED] Theoretical Derivation of Benford's Law

I have a question regarding a theoretical derivation of benford's law, the method used to derive it rests on the invariance of scale, first we transformed the x=1..10 initial domain into the y=c..10c getting

$\displaystyle f_{y}=1/c*f_{x}(y/c)$

Transformation was y = cx

And then the next transformation was a movement of the number back to 1..10 by saying that z=y if y < 10 or z=y/10 if y>=10, i then have the piecewise probability function of z as

$\displaystyle f_{z}=1/c*f_{x}(z/c)$ for y<10 or equiv x<10/c

and

$\displaystyle f_{z}=10/c*f_{x}(10z/c)$ for y>10 or equiv x>=10/c

The next step then says that based on the invariance of scale, fz is equal to fx but i can't understand how we are meant to derive the distribution of the function from this, knowing the distribution beforehand is log(1+1/x) helps and i can see how the invariance works but attempting to find c without first knowing that the distribution is fixed is very confusing

kindly explain how to compare the fx and fz

Hi,

I had a very similar question asked in a high school assignment in year 12 and was unable to do it. I lost a good deal of marks for that. I was able to do to the point you explained but then had to derive a form for fx. Though at that point I thought that there must be some information missing in the question but now seeing your question I think I was not clear about the question itself. Could you try and explain how you got the log expression?

Thanks a lot!