2.16 ( ) www Consider two random variables x1 and x2 having Gaussian distributions with means 1, 2 and precisions 1, 2 respectively. Derive an

2.16 ( ) www Consider two random variables x1 and x2 having Gaussian distributions with means μ1, μ2 and precisions τ1, τ2 respectively. Derive an expression for the differential entropy of the variable x = x1 + x2. To do this, first find the distribution of x by using the relation p(x) =

 ∞

−∞

p(x|x2)p(x2) dx2 (2.284)

and completing the square in the exponent. Then observe that this represents the convolution of two Gaussian distributions, which itself will be Gaussian, and finally make use of the result (1.110) for the entropy of the univariate Gaussian.