Scale Invariance
Change variables and use fact that the integral of the distribution
is $1$ to give result.
The only scale invariant functions are $B/x$ for any constant
$B$
These cannot be distributions on non-negative numbers because the
integral on $[0, \infty]$ is '$log(0)$'
We can 'regularise' and integrate from $[a, \infty]$ : on this
range $1/(log(a) x)$ is a distribution.
The probabilities are equal to
| First digit |
Probability |
| 1 |
0.301 |
| 2 |
0.176 |
| 3 |
0.125 |
| 4 |
0.097 |
| 5 |
0.079 |
| 6 |
0.067 |
| 7 |
0.058 |
| 8 |
0.051 |
| 9 |
0.046 |