# Inequality of logarithm

Given that the probability distribution is:

\begin{equation*} X \sim \begin{pmatrix} x_1 & x_2 & x_3 & \dots & x_N \\ p_1 & p_2 & p_3 & \dots & p_N \\ \end{pmatrix} \end{equation*}

with $p_1 \leq p_2 \leq \dots \leq p_N$. Prove that:

$$-\sum_{i=1}^{N}p_i\log_2 p_i \geq 2(1-p_N)$$

The hint in the exercice is to use $$-\sum_{i=1}^{N}p_i\ln p_i \geq (1-p_N)$$

for $p_N \geq 0.5$, and

$$-\sum_{i=1}^{N}p_i\log_2 p_i \geq -\log_2p_N$$

for $p_N \leq 0.5$. But I didn't understood very well this hint.

• What is the matrix notation? Does it describe the probability of each element? – BlackMath Aug 31 '18 at 0:26
• Is this saying that $\sum_{i=1}^N p_i=1$? Or are the $p_i$ supposed to be primes? – Clayton Aug 31 '18 at 0:30
• Is this a Dirichlet distribution notation? en.wikipedia.org/wiki/Dirichlet_distribution – Maxtron Aug 31 '18 at 1:24
• I edit the post, its a PMF – Felipe Aug 31 '18 at 2:50