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# Entropy

Entropy is a measure of the average uncertainty about what the value of a random variable is before observing it. Entropy is measured in bits.

An entropy of H bits means that in order to provide information about the value of the as yet unobserved random variable, it will require, on the average, an H bit message. For example, an H bit message specifies a choice of 1 out of 2H possibilities.

One way to explain the meaning of the H bit message is by the following game played between person A and person B. Person A samples at random a value v of the random variable X. Person B knows what the probability is of random variable X taking any of its values, but does not know the value v that person A has sampled. If person B were to use his knowledge of the probability function of random variable X in the most effective way possible, it would take person B, on the average, 2H guesses to correctly guess the value v that person B had sampled.

If P denotes the probability function of a discrete random variable X which takes possible values {x1,...,xN} and H(X) denotes the entropy of the random variable X, then the entropy of the random variable X is minus the expected value of log to the base 2 of P(X)

H(X) = -E[log_2 P(X)]= - ΣNn=1 P(xn) log_2 P(xn}) .