I would need a probability distribution which reflects the following user input:
- Value 0.5...9.5
- Belief into that value
The higher the belief, the more the distribution is Dirac-like. The lower the belief, the more the distribution is flat. However, the distribution should have a finite support. In this concrete case, non-zero values only exist in the range 0...10 exclusive. If the range were 0...infinity, I could use a lognormal distribution, but it is not. I do not want to use a truncated normal distribution, because it does not have zero values at the extreme points of its support.
Here is a chart of two lognormal distributions and also the green plot representing a lognormal swapped around 5:
I would like the searched distribution to behave more-or-less similarly.