Exercise 3.4: Entropy and Constant Shifts

chapter 3

(a)

$$\begin{array}{}\text{(1)}& \mathbb{H}(e^{\lambda X})& =e^{\lambda c}\{\mathbb{E}[\lambda (X+c)e^{\lambda X}]-\mathbb{E}[e^{\lambda X}]\log \mathbb{E}[e^{\lambda (X+c)}]\}\text{(2)}& & =e^{\lambda c}\{\mathbb{E}[\lambda X{e}^{\lambda X}]+\lambda c\mathbb{E}[e^{\lambda X}]-\mathbb{E}[e^{\lambda X}]\log \mathbb{E}[e^{\lambda X}]-\lambda c\mathbb{E}[e^{\lambda X}]\}\text{(3)}& & =e^{\lambda c}\mathbb{H}(e^{\lambda X}).\end{array}$$

(b)

This follows by the MGF identity ${\phi }_{X+c}(\lambda )=e^{\lambda c}{\phi }_X(\lambda )$ combined with (a).