Variance of a special random walk

Question

I am trying to find the variance of the following special random walk:

Suppose that $U=(U_1,U_2,...)$ is a sequence of independent random variables, each taking values $u$ (for up) and $d$ (for down) with probability $p=\frac{1}{2}$. Let $X=(X_0,X_1,X_2,...)$ be the partial sum process associated with $U$ so that $$X_n=\sum\limits_{i=1}^n U_i, \hspace{1cm} n\in\mathbb{N}$$

The mean of this process is $$\mathbb{E}(X_n)=n\cdot p\cdot(u+d)$$

My question
What is the variance $var(X_n)$?

Answer 1

If the $U_n$ are i.i.d. then $$\mathrm{Var}(X_n) = \mathrm{Var}\left(\sum_{i=1}^n U_i\right) = \sum_{i=1}^n\mathrm{Var}(U_i) = n\mathrm{Var}(U_1).$$