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1.7. Problem*. Let X be a random variable with support contained in the positive real numbers. Prove each of the following inequalities: (a) VEX] Z

1.7. Problem*. Let X be a random variable with support contained in the positive real numbers. Prove each of the following inequalities: (a) VEX] Z E VX. (b) E [ X] > EX (c) E[-log X] 2 - log(E[X]) (d) E [log ( ) ] > log E X]

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