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Prove that the quantity S= -kΣP li(P) r=1 is a maximum when P, = 1/n for all r. Do this by showing that S - kIn n < 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 E- P, = 1 in an appropriate place. This is a problem which is easier not using calculus because of the probability constraint.]

Prove that the quantity S= -kΣP li(P) r=1 is a maximum when P, = 1/n for all r. Do this by showing that S - kIn n < 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 E- P, = 1 in an appropriate place. This is a problem which is easier not using calculus because of the probability constraint.]

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Prove that the quantity S= -kΣP li(P) r=1 is a maximum when P, = 1/n for all r. Do this by showing that S - kIn n < 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 E- P, = 1 in an appropriate place. This is a problem which is easier not using calculus because of the probability constraint.]
Transcribed Image Text:Prove that the quantity S= -kΣP li(P) r=1 is a maximum when P, = 1/n for all r. Do this by showing that S - kIn n < 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 E- P, = 1 in an appropriate place. This is a problem which is easier not using calculus because of the probability constraint.]
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