Suppose that \( X \) is a discrete random variable on the set \( \{ a_1, a_2, \dots \} \) with \( P(X=a_i) = p_i \). Define the discrete entropy

\[

H(X) = -\sum_{n=1}^{\infty} p_i \log p_i.

\]

Find constants \( C_1, C_2 \geq 0 \) such that

\[

e^{2H(X)} \leq C_1 Var(X) + C_2

\]

holds for any \( X \).

The best solution was submitted by 길현준 (2018학번). Congratulations!

Here is his solution of problem 2019-09.

Alternative solutions were submitted by 최백규 (생명과학과 2016학번, +3). Incomplete solutions were submitted by, 이정환 (수리과학과 2015학번, +2), 채지석 (수리과학과 2016학번, +2).

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