Show that $M_n = X_n^2 - n$ is a martingale Suppose $X_n$ is a symmetric random walk on $\mathbb{Z}$. To show that it is a martingale I need to show
$$
\mathbb{E}[M_{n+1}|X_{0:n}] = M_n
$$

$$
\begin{align}
\mathbb{E}[M_{n+1}|X_{0:n}] &amp;= \mathbb{E}[X_{n+1}^2 - (n+1)|X_{0:n}]\\
&amp;= \mathbb{E}[X_{n+1}^2|X_{0:n}] - (n+1)
\end{align}
$$

I'm stuck here.
Hint: $$X_{n+1}^2 = X_n^2 + (X_{n+1}-X_n)^2 + 2X_n(X_{n+1} - X_n)$$
Assume that the question without the -n is a martingale. The latter is easy to prove M (n) =X^2 (n). The expectation of X^2 (n+1) given the past information up to n is E(X2n) = n because it is like a wiener process that is Normal with mean zero and variance the time  (t) then the process you have has an expectation of zero and variance of 2(....)by the use of the definition of expectation and variance