(a) Using the $P$ representation of the density operator for a chaotic field, show that
$$
\left\langle e^{\mu^* a^{\dagger}-\mu a}\right\rangle=e^{-|\mu|^2(\langle n\rangle+1 / 2)}, \quad\langle n\rangle=\left\langle a^{\dagger} a\right\rangle
$$
(Hint: use (3.6.14) to put the operator $e^{\mu^* a^{\dagger}-\mu a}$ in normal order.) (b) Using (3.7.22), show that the probability distribution for the single-mode electric field $E=i C\left(a-a^{\dagger}\right)$ for chaotic light is
$$
P(\mathcal{E})=\sqrt{\frac{1}{2 \pi\left\langle E^2\right\rangle}} e^{-\mathcal{E}^2 / 2\left\langle E^2\right\rangle}
$$
(c) Consider a squeezed state for which $\Delta X_i<1 / 2, i=1$ or 2 , with $X_i$ defined by (3.6.77). Show that $P(\alpha)$ cannot be a positive-definite distribution function.