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How to calculate the entropy of two random variables?
Mutual information, entropy For any two random variables X and Y we have I(X;Y) 0 with equality if and only if X and Y are independent. Proof: I(X;Y) = D(p(x;y)jjp(x)p(y)) 0 with equality if and only if p(x;y) = p(x)p(y), ie X and Y are independent. The entropy is bounded by the logarithm of the alphabet size H(X) log jXj Proof: Let u(x) = 1 jXj
What is the expectation of a Cauchy random variable?
A Cauchy random variable takes a value in (−∞,∞) with the fol- lowing symmetric and bell-shaped density function. f(x) = 1 π[1+(x−µ)2] The expectation of Bernoulli random variable implies that since an indicator function of a random variable is a Bernoulli random variable, its expectation equals the probability.
Is the expectation of a random variable a linear operator?
In particular, the following theorem shows that expectation preserves the inequality and is a linear operator. Theorem 1 (Expectation) Let X and Y be random variables with finite expectations. 1. If g(x) ≥ h(x) for all x ∈ R, then E[g(X)] ≥ E[h(X)].
What is the joint entropy of a variable?
Joint and conditional entropy. The joint entropy of a pair of discrete random variables (X;Y) with joint probability mass function p(x;y) is de ned by H(X;Y) = X.
Shannon proposed the next function to measure the information of each possible event ( A) in a discrete source as: where P ( A) corresponds to the probability associated to the occurrence of the event A. In Fig. 7.26 we sketch the idea of the information by Shannon.
How is the Shannon entropy used in molecular reconstruction?
In the context of molecular reconstruction, the Shannon entropy is defined as H = − n ∑ i = 1x i · logxi, and solving the reconstruction is posed as an optimization problem, where xi is the mass fraction for each compound that constitute the mixture ( Hudebine et al., 2011 ).