grandes-ecoles 2023 Q6

grandes-ecoles · France · x-ens-maths__psi Not Maths

Let $q = (2^{-|c(x)|})_{x \in X}$ and $X$ a random variable taking values in $X$ with distribution $p$.
(a) Verify that $\ln(2) E(|c(X)|) = -\sum_{x \in X} p_x \ln(q_x)$.
(b) Deduce that $E(|c(X)|) \geq \frac{H(p)}{\ln(2)}$. (Hint: One may try to express $\ln(2) E(|c(X)|)$ in terms of $H(p)$ and $\mathrm{KL}(p, q)$)

Let $q = (2^{-|c(x)|})_{x \in X}$ and $X$ a random variable taking values in $X$ with distribution $p$.\\
(a) Verify that $\ln(2) E(|c(X)|) = -\sum_{x \in X} p_x \ln(q_x)$.\\
(b) Deduce that $E(|c(X)|) \geq \frac{H(p)}{\ln(2)}$.\\
(Hint: One may try to express $\ln(2) E(|c(X)|)$ in terms of $H(p)$ and $\mathrm{KL}(p, q)$)

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