Let $X$ be a real-valued random variable. Define the characteristic function $\phi_{X}$. Show that $\phi_{X}(u) \in \mathbb{R}$ for all $u \in \mathbb{R}$ if and only if $X$ and $-X$ have the same distribution.

For parts (a) and (b) below, let $X$ and $Y$ be independent and identically distributed random variables.

(a) Show that $X=Y$ almost surely implies that $X$ is almost surely constant.

(b) Suppose that there exists $\varepsilon>0$ such that $\left|\phi_{X}(u)\right|=1$ for all $|u|<\varepsilon$. Calculate $\phi_{X-Y}$ to show that $\mathbb{E}(1-\cos (u(X-Y)))=0$ for all $|u|<\varepsilon$, and conclude that $X$ is almost surely constant.

(c) Let $X, Y$, and $Z$ be independent $\mathrm{N}(0,1)$ random variables. Calculate the characteristic function of $\eta=X Y-Z$, given that $\phi_{X}(u)=e^{-u^{2} / 2}$.