Let $X_n$, $n=1,2,...$ be independent random variables. $P(X_n=1)=\frac{1}{2}=P(X_n=-1)$
$S_n=\sum_1^nX_k$, $T=inf[n:S_n=n-2008]$
a) Prove that $P(\exists n\in N:S_n=n-2008)=1$
b) Prove $ET\leq2008$
My attempt is obviously wrong, but let me show you.
$T$ is a stopping moment, ${n\wedge T}$ is also a stopping moment, S is a martingale
$$ES_{n\wedge T}=\Bbb E[S_T1_{\{T \le n\}}]+\Bbb E[S_n1_{T> n}]=0.$$
$$S_T=n-2008$$ which goes to infinity as n goes to infinity, so my result would be the opposite of thesis a), namely $P(T<\infty)=0$
EDIT:
my new idea is to form another variable $Z_n=S_n-n+2008$, this would be a supermartingale.