Inequality of logarithm

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Given that the probability distribution is:



beginequation*
X sim
beginpmatrix
x_1 & x_2 & x_3 & dots & x_N \
p_1 & p_2 & p_3 & dots & p_N \
endpmatrix
endequation*



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



$$ -sum_i=1^Np_ilog_2 p_i geq 2(1-p_N) $$



The hint in the exercice is to use
$$ -sum_i=1^Np_iln p_i geq (1-p_N) $$



for $p_N geq 0.5$, and



$$-sum_i=1^Np_ilog_2 p_i geq -log_2p_N $$



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






inequality







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  • What is the matrix notation? Does it describe the probability of each element?
    – BlackMath
    Aug 31 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 at 0:30










  • Is this a Dirichlet distribution notation? en.wikipedia.org/wiki/Dirichlet_distribution
    – Maxtron
    Aug 31 at 1:24










  • I edit the post, its a PMF
    – Felipe
    Aug 31 at 2:50














up vote
2
down vote

favorite












Given that the probability distribution is:



beginequation*
X sim
beginpmatrix
x_1 & x_2 & x_3 & dots & x_N \
p_1 & p_2 & p_3 & dots & p_N \
endpmatrix
endequation*



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



$$ -sum_i=1^Np_ilog_2 p_i geq 2(1-p_N) $$



The hint in the exercice is to use
$$ -sum_i=1^Np_iln p_i geq (1-p_N) $$



for $p_N geq 0.5$, and



$$-sum_i=1^Np_ilog_2 p_i geq -log_2p_N $$



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






inequality







share|cite|improve this question























  • What is the matrix notation? Does it describe the probability of each element?
    – BlackMath
    Aug 31 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 at 0:30










  • Is this a Dirichlet distribution notation? en.wikipedia.org/wiki/Dirichlet_distribution
    – Maxtron
    Aug 31 at 1:24










  • I edit the post, its a PMF
    – Felipe
    Aug 31 at 2:50












up vote
2
down vote

favorite









up vote
2
down vote

favorite











Given that the probability distribution is:



beginequation*
X sim
beginpmatrix
x_1 & x_2 & x_3 & dots & x_N \
p_1 & p_2 & p_3 & dots & p_N \
endpmatrix
endequation*



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



$$ -sum_i=1^Np_ilog_2 p_i geq 2(1-p_N) $$



The hint in the exercice is to use
$$ -sum_i=1^Np_iln p_i geq (1-p_N) $$



for $p_N geq 0.5$, and



$$-sum_i=1^Np_ilog_2 p_i geq -log_2p_N $$



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






inequality







share|cite|improve this question















Given that the probability distribution is:



beginequation*
X sim
beginpmatrix
x_1 & x_2 & x_3 & dots & x_N \
p_1 & p_2 & p_3 & dots & p_N \
endpmatrix
endequation*



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



$$ -sum_i=1^Np_ilog_2 p_i geq 2(1-p_N) $$



The hint in the exercice is to use
$$ -sum_i=1^Np_iln p_i geq (1-p_N) $$



for $p_N geq 0.5$, and



$$-sum_i=1^Np_ilog_2 p_i geq -log_2p_N $$



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






inequality




inequality






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Aug 31 at 16:55

























asked Aug 31 at 0:18









Felipe

797




797











  • What is the matrix notation? Does it describe the probability of each element?
    – BlackMath
    Aug 31 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 at 0:30










  • Is this a Dirichlet distribution notation? en.wikipedia.org/wiki/Dirichlet_distribution
    – Maxtron
    Aug 31 at 1:24










  • I edit the post, its a PMF
    – Felipe
    Aug 31 at 2:50
















  • What is the matrix notation? Does it describe the probability of each element?
    – BlackMath
    Aug 31 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 at 0:30










  • Is this a Dirichlet distribution notation? en.wikipedia.org/wiki/Dirichlet_distribution
    – Maxtron
    Aug 31 at 1:24










  • I edit the post, its a PMF
    – Felipe
    Aug 31 at 2:50















What is the matrix notation? Does it describe the probability of each element?
– BlackMath
Aug 31 at 0:26




What is the matrix notation? Does it describe the probability of each element?
– BlackMath
Aug 31 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 at 0:30




Is this saying that $sum_i=1^N p_i=1$? Or are the $p_i$ supposed to be primes?
– Clayton
Aug 31 at 0:30












Is this a Dirichlet distribution notation? en.wikipedia.org/wiki/Dirichlet_distribution
– Maxtron
Aug 31 at 1:24




Is this a Dirichlet distribution notation? en.wikipedia.org/wiki/Dirichlet_distribution
– Maxtron
Aug 31 at 1:24












I edit the post, its a PMF
– Felipe
Aug 31 at 2:50




I edit the post, its a PMF
– Felipe
Aug 31 at 2:50















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