note that because the result does not depend on other than through the variance. Combining the two results yields with equality when following from the properties of Kullback–Leibler divergence. Alternative proof This result may also be demonstrated using the calculus of variations. A Lagrangian function with two Lagrangian multipliers may be defined as: where g(x) is some function with mean μ. When the entropy of g(x) is at a maximum and the constraint equations, which consist of the normalization condition and the requirement of fixed variance , are both satisfied, then a small variation δg(x) about g(x) will produce a variation δL about L which is equal to zero: Since this must hold for any small δg(x), the term in brackets must be zero, and solving for g(x) yields: Using the constraint equations to solve for λ0 and λ yields the normal distribution: Example: Exponential distribution Let be an exponentially distributed random variable with parameter , that is, with probability density function Its differential entropy is then Here, was used rather than to make it explicit that the logarithm was taken to base e, to simplify the calculation. Relation to estimator error The differential entropy yields a lower bound on the expected squared error of an estimator. For any random variable and estimator the following holds: with equality if and only if is a Gaussian random variable and is the mean of . Differential entropies for various distributions In the table below is the gamma function, is the digamma function, is the beta function, and γE is Euler's constant. Many of the differential entropies are from. Variants As described above, differential entropy does not share all properties of discrete entropy. For example, the differential entropy can be negative; also it is not invariant under continuous coordinate transformations. Edwin Thompson Jaynes showed in fact that the expression above is not the correct limit of the expression for a finite set of probabilities. A modification of differential entropy adds an invariant measure factor to correct this, (see limiting density of discrete points). If is further constrained to be a probability density, the resulting notion is called relative entropy in information theory: The definition of differential entropy above can be obtained by partitioning the range of into bins of length with associated sample points within the bins, for Riemann integrable. This gives a quantized version of , defined by if . Then the