(a) Let $\Omega=(0,1), \mathcal{F}=\mathcal{B}((0,1))$ be the Borel $\sigma$-field and let $\mathbb{P}$ be Lebesgue measure on $(\Omega, \mathcal{F})$. What is the distribution of the random variable $Z$, where $Z(\omega)=2 \omega-1$ ?

Let $\omega=\sum_{n=1}^{\infty} 2^{-n} R_{n}(\omega)$ be the binary expansion of the point $\omega \in \Omega$ and set $U(\omega)=\sum_{n \text { odd }} 2^{-n} Q_{n}(\omega)$, where $Q_{n}(\omega)=2 R_{n}(\omega)-1$. Find a random variable $V$ independent of $U$ such that $U$ and $V$ are identically distributed and $U+\frac{1}{2} V$ is uniformly distributed on $(-1,1)$.

(b) Now suppose that on some probability triple $X$ and $Y$ are independent, identicallydistributed random variables such that $X+\frac{1}{2} Y$ is uniformly distributed on $(-1,1)$.

Let $\phi$ be the characteristic function of $X$. Calculate $\phi(t) / \phi(t / 4)$. Show that the distribution of $X$ must be the same as the distribution of the random variable $U$ in (a).