Max diff entropy(Info. theory)

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If X is continuous r.v. and has pdf only in the positive real axis with E[X]=\alpha, E[X^2]=\beta, what is max diff. entropy of X?
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To solve this, I first used the units to work out that a= m* a/m, i.e. t=z/λ. This would allow you to determine the time duration within an interval section by section and then add this to the previous ones to obtain the age of the respective layer. However, this would require a constant thickness per year for each interval. However, since this is most likely not the case, my next consideration was that the age must be the integral of a 1/λ(z) function, which I cannot model.
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