2.II .9 F. 9 \mathrm{~F} \quad

Probability | Part IA, 2005

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

E[eiX]E[cosX]+iE[sinX]\mathbb{E}\left[e^{i X}\right] \equiv \mathbb{E}[\cos X]+i \mathbb{E}[\sin X]

Consider a second real-valued random variable YY, independent of XX. Show that

E[ei(X+Y)]=E[eiX]E[eiY]\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£ 1 with equal probability 1/21 / 2. Let WnW_{n} denote your wealth after the nthn^{t h} game. For a fixed real number uu, compute ϕ(u)\phi(u) defined by

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

Verify that the result is real-valued.

Show that for nn even,

P[Wn=0]=γ0π/2[cosu]ndu\mathbb{P}\left[W_{n}=0\right]=\gamma \int_{0}^{\pi / 2}[\cos u]^{n} d u

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

Typos? Please submit corrections to this page on GitHub.