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2. Let X be a Bernoulli random variable with probability of X -1 being a. a) Write down the probability mass function p(X) of X in terms of a. Mark the range of a (b) Find the mean value mx(a) EX] of X, as a function of a (c) Find the variance σ剤a) IX-mx)2) of X, as a function of a. (d) Consider another random variable Y as a function of X: Y = g(X) =-log p(X) where the binary logarithm has base 2. Find the mean value my(a) EY] of Y as a function of a. We now define a function of X as H(X: a) = my(a) = El-log p(X)]. It is termed the entropy of X (e) Plot both H(X; ) and ơ3 (a) versus a. Mark all critical parameters clearly. (f) Find the maximum and minimum points of H(X;a). For each maximum or minimum point, write down the corresponding values of a, H(Xa*)-my(a*), and the variance σ (a*) respectively

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