Define a Sigmoid relation between two variables

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I have the variable A: 'Probability of X' which takes the range [0, 1]. I need to define a second variable B: 'Risk rate' with range [0, 1] as a function of A. The requirements are:

  • If A = 0 then B = 0,

  • If A = max(A) then B = 1

  • They must follow a classic Sigmoid function.

I would really appreciate any help or suggestion.