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 $var(X_n)$?

  • 1
    $\begingroup$ OK. What about the variance of $X_1$? $\endgroup$ – Did May 30 '15 at 16:51
  • $\begingroup$ Re the Edit: The additivity of the variances of sums is guaranteed as soon as independence holds, for every distribution, whether skewed or not. $\endgroup$ – Did May 31 '15 at 9:15
  • $\begingroup$ Impossible to say since you do not show said calculations but only the (wrong) result you arrive at. $\endgroup$ – Did May 31 '15 at 10:31
  • $\begingroup$ What do you call var(a,b) and var(a,b,c,d) for real numbers (a,b,c,d)? Never saw the notation. $\endgroup$ – Did May 31 '15 at 10:38
  • $\begingroup$ @Did: You asked for the bigger context of this problem and I posted another question here: math.stackexchange.com/questions/1309230/… - This time I really tried hard to give all the necessary information, let's see if this is sufficient for your critical eye ;-) $\endgroup$ – vonjd Jun 2 '15 at 13:15

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

  • $\begingroup$ Thanks for catching my typo @Did :) $\endgroup$ – Math1000 May 30 '15 at 18:19

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