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Maximal number of independent random variables

  1. Feb 10, 2008 #1
    Hi all,

    assume we have a sample space with 2^n points. (it size is 2^n for some natural n)
    I need to prove that the maximal number of independent binary (indicator) random variables (which are not trivial, i.e. constant) is n...



    Thnks,
    Pitter
     
  2. jcsd
  3. Feb 10, 2008 #2

    mathman

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    Have you tried mathematical inuction on n. It seems pretty straightforward.
     
  4. Feb 10, 2008 #3

    EnumaElish

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    I am not sure that I understand.

    Say I have 4 obs with 3 independent variables:

    y x1 x2 x3
    10 1 0 1
    11 1 0 0
    12 0 1 0
    20 0 1 1

    I can estimate the model y = b1 x1+ b2 x2 + b3 x3 + u with the following results:
    b1 = 8.75
    b2 = 14.25
    b3 = 3.5
     
  5. Feb 10, 2008 #4
    I've tried induction,

    but I still don't get it...

    I don't really know how to algebraically define
    dependent random variables (dependent by their random variable function)
     
  6. Feb 10, 2008 #5

    EnumaElish

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    I am not convinced that the statement is true. Did you see my example?

    n = 3
    2^n = 8

    Since I can uniquely and independently estimate a 3-variable model with 4 observations, I can surely estimate, say, a 4-variable model with 8 observations. Yet the statement implies that with 8 observations the maximal number of variables is 3.

    Another example with 2^3 = 8 observations and 4 independent binary variables:
    x1 x2 x3 x4
    1 0 1 0
    1 0 0 1
    0 1 0 1
    0 1 1 1
    1 0 1 1
    1 0 0 1
    0 1 0 1
    0 1 1 0
     
    Last edited: Feb 10, 2008
  7. Feb 10, 2008 #6

    EnumaElish

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    Random variables X and Y are dependent if Prob{X = x and Y = y} [itex]\ne[/itex] Prob{X=x}Prob{Y=y}, or equivalently, Prob{X < x and Y < y} [itex]\ne[/itex] Prob{X<x}Prob{Y<y}.

    Alternatively, variables x1, x2, x3 and x4 are linearly dependent if there exists a1, a2, a3 such that x4 = a1 x1 + a2 x2 + a3 x3. Equivalently, they are lin. dep. if there exists a1, a2, a3 and a4, not all equal to zero, such that a1 x1 + a2 x2 + a3 x3 + a4 x4 = 0.

    Neither of my examples above is a case of linear dependence.
     
    Last edited: Feb 10, 2008
  8. Feb 11, 2008 #7
    thanks EnumaElish,

    I think you're right,
    and if looking at it that way (as vector space) the maximum number of independent binary random variables above 2^n points in the sample space, is 2^n....and not n.
     
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