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Distribution of the decimals of a random number

  1. Jul 10, 2010 #1
    1. The problem statement, all variables and given/known data

    Let U = 0.X1X2X3... be a random number in (0,1].

    1) Find the distribution of every decimal digit Xi, i = 0,1,2...

    2) Show that they are independent of each other

    3. The attempt at a solution

    I could use a hint for N°2. I have an idea, but I think it's wrong:

    If X1 and X2 are independent, then P(X2 = x2 (intersection) X1 = x1) = P(X2 = x2)*P(X1 = x1).

    Since P(X2 = x2 (intersection) X1 = x1) = P(0.x1x2... < U <= 0.x1(x2+1)...) = 0.x1(x2+1) - 0.x1x2 = 0.01 = 0.1*0.1 = P(X2 = x2) * P(X1 = x1) , that should prove it. Then, I could invoke the induction principle. But something sounds off.

    Is this correct?
     
  2. jcsd
  3. Jul 11, 2010 #2
    Anybody?
     
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