Exercise 3.4: Entropy and Constant Shifts

chapter 3

(a)

\begin{align} \mathbb{H} (e^{\lambda X}) &= e^{\lambda c} \left\lbrace \E [\lambda (X + c) e^{\lambda X}] - \E [e^{\lambda X}] \log \E [e^{\lambda (X + c)}] \right\rbrace \newline &= e^{\lambda c} \left\lbrace \E [\lambda X e^{\lambda X}] + \lambda c \E [e^{\lambda X}] - \E [e^{\lambda X}] \log \E [e^{\lambda X}] - \lambda c \E [e^{\lambda X}] \right\rbrace \newline &= e^{\lambda c} \mathbb{H} (e^{\lambda X}) \, . \end{align}

(b)

This follows by the MGF identity \(\varphi_{X + c} (\lambda) = e^{\lambda c} \varphi_X (\lambda)\) combined with (a).