For a continuous random variable \(X\) with distribution \(p_{X}(x)\), the uncertainty is measured by the so-called differential entropy \(h(X)\) determined as (Cover \& Thomas, 2006)

\[H(X)=-\int_{x} p_{X}(x) \log _{b} p_{X}(x) \mathrm{d} x .\]

Determine the differential entropy of the random variables characterized by:

(a) continuous uniform distribution \(u_{X, \mathrm{c}}(x)\) given in Equation (1.207)

(b) the Gaussian distribution \(\phi_{X}(x)\) given in Equation (1.209).

Transcribed Image Text:

1 ox(x) = e-(x-)/22 2x02 (1.209)