Let X be a random variable that takes on 5 possible values with respective probabilities .35, .2, .2, .2, and .05. Also, let Y be a random variable that takes on 5 possible values with respective probabilities .05, .35, .1, .15, and .35.
(a) Show that H(X) > H(Y).
(b) Using the result of Problem 1, give an intuitive explanation for the preceding inequality.
Problem 1
Prove that if X can take on any of n possible values with respective probabilities P1,..., Pn,then H(X) is maximized when Pi = 1/n, i = 1,..., n. What is H(X) equal to in this case?
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