Given a real-valued random variable $X$, we define $\mathbb{E}\left[e^{i X}\right]$ by

$$\mathbb{E}\left[e^{i X}\right] \equiv \mathbb{E}[\cos X]+i \mathbb{E}[\sin X]$$

Consider a second real-valued random variable $Y$, independent of $X$. Show that

$$\mathbb{E}\left[e^{i(X+Y)}\right]=\mathbb{E}\left[e^{i X}\right] \mathbb{E}\left[e^{i Y}\right]$$

You gamble in a fair casino that offers you unlimited credit despite your initial wealth of 0 . At every game your wealth increases or decreases by $£ 1$ with equal probability $1 / 2$. Let $W_{n}$ denote your wealth after the $n^{t h}$ game. For a fixed real number $u$, compute $\phi(u)$ defined by

$$\phi(u)=\mathbb{E}\left[e^{i u W_{n}}\right]$$

Verify that the result is real-valued.

Show that for $n$ even,

$$\mathbb{P}\left[W_{n}=0\right]=\gamma \int_{0}^{\pi / 2}[\cos u]^{n} d u$$

for some constant $\gamma$, which you should determine. What is $\mathbb{P}\left[W_{n}=0\right]$ for $n$ odd?