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The image below shows two identically shaped (Normal) distributions with the second only different by its mean. If I calculate the differential entropy of both separately, would the entropies of the two distributions be equal to one another? If so, doesn't this defeat the whole intuition behind statistical inference for real continuous data?

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Yes, if a variable is sampled from the standard Normal distribution, and we also create a carbon copy of that variable, except whose mean is shifted up by $x$ by adding $+2$ to all observations in the carbon copy, then the entropy of the original variable, and the entropy of its copy with the shifted-mean, will be identical.

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