Variance in Random Walk

  • Thread starter grad
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  • #1
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  • #2
Just write down the definition of [itex]S_n[/itex] and you will be able to answer your question yourself.
 
  • #3
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Var(Sn) = E(Sn2) = E(Z12 + Z22 + Z32 + ... + Zn2) =* E(Z12) + E(Z22) + ... + E(Zn2) = 1 + 1 + ... + 1 (n times) = n

*variables are independent and uncorrelated

Is this correct then?
 
  • #4
This is almost correct. [itex]S_n[/itex] is defined to be [itex]Z_1+\ldots +Z_n[/itex], where the [itex]Z_i[/itex] are independent (or at least uncorrelated) with mean zero and variance one. It follows that
[tex]
S_n^2 = \sum_{i,j=1}^n{Z_i Z_j}
[/tex]
and not, as you wrote,
[tex]
S_n^2 = \sum_{i=1}^n{Z_i^2}
[/tex]

However, using independence of the [itex]Z_i[/itex] you can still do a similar computation to prove [tex]\mathbb{E}\left[S_n^2\right]=n[/tex].
 
  • #5
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Thank you!
 
  • #6
You're welcome:smile:
 

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