Assume $X_n$ is an iid Gaussian random process with zero mean and variance $\sigma^2$, and $U_n$ be an iid binary random process with $P_r\{ U_{n}=1\}=P_r\{U_n=-1\}=0.5$, and $\{U_n\}$ is independent of $\{X_n\}$, now let $Q_n=X_n+U_n$,and $B_n=X_n+U_0$,what is the mean of $Q_n$ and $B_n$ ?
In my opinion
$E[Q_n]=E[X_n+U_n]=E[X_n]+E[U_n]=0+(1 \times 0.5+-1 \times 0.5)=0$
$E[B_n]=E[X_n+U_0]=E[X_n]+E[U_0]=0+(\frac{1}{2} \times 1 \times 0.5+\frac{1}{2} \times -1 \times 0.5)=0$
Actually,i am not sure about that the $E[U_0]$ is $0$ ,if wrong,can anyone tell me how to calculate it?
$U_n$ are iid. It implies that $U_1$, $U_2$, ... have all the same expectation.
So actually, $E[Q_n]=E[B_n]=E[Q_1]=E[B_1]= 0$. The expectations do not depend on $n$ at all. Your calculations are correct.