-4
$\begingroup$

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 $var(X_n)$?

$\endgroup$
  • 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
2
$\begingroup$

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

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.