en.wikipedia.org/wiki/Maximum_entropy_probability_distribution

In the table, this expression appears in the Support column for the logistic distribution. That is mathematically incorrect.

The logistic distribution is a continuous distribution defined for every real value. Standard references describe it as being supported over the set of real numbers / for (y \in \mathbb{R}). By contrast, the notation ({-\infty, \infty}) denotes a set containing only two elements, negative infinity and positive infinity, not the whole real line.

So the correct support should be written as (\mathbb{R}) or ((-\infty, \infty)), not ({-\infty, \infty}).

The preceding inequality,

[
H(q) \ge \alpha H(p) + (1-\alpha)H(p')
]

is the standard concavity property of entropy for the mixture (q = \alpha p + (1-\alpha)p'). But the sentence quoted goes further and concludes that for the same fixed mixture (q) one gets

[
H(q) \ge H(p),; H(p').
]

That conclusion does not follow, and it is false in general.

A concrete counterexample uses the discrete entropy formula (H(X)=-\sum_i p_i \log p_i):

• Let (p=(1/2,1/2)), so (H(p)=\ln 2 \approx 0.693).
• Let (p'=(9/10,1/10)).
• Take (\alpha=1/2). Then
  [
  q = \tfrac12 p + \tfrac12 p' = (0.7,0.3).
  ]
• Its entropy is
  [
  H(q)=-(0.7\ln 0.7 + 0.3\ln 0.3) \approx 0.611.
  ]

So in this example,

[
H(q) \approx 0.611 < 0.693 \approx H(p),
]

which directly contradicts the claim that (H(q) \ge H(p)) and (H(q) \ge H(p')).

What is true is only the concavity inequality above; it does not imply that a nontrivial mixture has entropy at least as large as each component individually.