How do I prove this? I'm thinking I should use Jensen's inequality somehow.
$$\sum_K p_k(1-p_k) \le -\sum_K p_k\log p_k$$
The assumption that $\sum_K p_k=1$ holds.
2026-03-26 17:32:55.1774546375
prove this inequality related to probability and information theory
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As explained in the comments, the inequality $$\sum_kp_k(1-p_k)\lt\sum_kp_k\log p_k$$ cannot hold since the LHS is nonnegative and the RHS is nonpositive.
To show that $$\sum_kp_k(1-p_k)\leqslant-\sum_kp_k\log p_k,$$ note that, for every positive $x$, $$1-x\leqslant-\log x.$$