Question

k is a constant, it can be positive or negative

3. Prove that the quantity s--k Σ P, In(R) r=1 is a maximum when P, = 1/n for all r. Do this by showing that S-kinn-0. Hint: The inequality In [1/(nP )] 1/(nP,) 1 will be helpful. TO arrive at an expression in which you can use this inequality, you will have to insert Ση_B-1 in an appropriate place. This is a problem which is easier not using calculus because of the probability constraint.]

Answer 1

S = -k sigma^n_r = 1 p_r log_2 (p_r), sigma^n_r = 1 p_r = 1 consider k log_2 n - S = k log_2 n sigma^n_r = 1 p_r + k sigma^n_r = 1 p_r log_2 (p_r) = k sigma^n_r = 1 p_r log_2 n + k sigma^n_r = 1 p_r log_2 (p_r) = k sigma^n_r = 1 p_r log_2 n + k sigma^n_r = 1 p_r log_2 (p_r) = k sigma^n_r = 1 p_r log_2 (p_r)/1/n -----(1) Next we will prove log_2 y lessthanorequalto y - 1/log_e 2 forall y elementof IR implies log_e (1 + x) lessthanorequalto x (changing the base) implies log_2 (1 + x) lessthanorequalto x/log_e 2 let y = x + 1, then log_2 y lessthanorequalto y -1/log_e 2 (proved) ---(11) consider (1) k sigma^n_r = 1 p_r log_2 p_r/1/n = -k sigma^n_r = 1 p_r log_2 1/n/p_r (consider y = 1/n/p_r in (1)) .