Show $M_n=X_1+X_2+...+X_n-n\theta$ is a martingale w.r.t ${X_n}$, given that $X_i$ are i.i.d. random variables with $\mathbb{E}[X_i]=\theta$
this is what I've done: $$\mathbb{E}[M_{n+1}|X_{\le n}]=\mathbb{E}[X_1+...+X_n+X_{n+i}-(n+1)\theta|X_{\le n}]$$ $$\mathbb{E}[X_1+X_2+...+X_n-n\theta|X_{\le n}]+\mathbb{E}[X_{n+1}-\theta|X_{\le n}]$$ $$M_n+\theta-\theta=M_n$$
is this correct?