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Math Help - Why is this distribution non-uniform?

  1. #1
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    Why is this distribution non-uniform?

    So, here is the situation. I'm taking a string of 5 integer digits and using a formula to convert that string of digits into a real-valued number.

    The basic formulation is this:

     \text{String} = abcde

     \text{Real} = (a+0.1\times b + 0.01 \times c + 0.001 \times d) \times 10^{\frac{e}{2}-2}

    So for example...

     \text{String} = 56796

     \text{Real} = (5+0.1\times 6 + 0.01 \times 7 + 0.001 \times 9) \times 10^{\frac{6}{2}-2} = 5.679 \times 10^{1} = 56.79

    Now, if I set up 10,000 such strings, and populate each string with 5 randomly (and uniformly) selected integers in the interval [0,9] , then I would expect to get a nice uniform distribution of reals on the interval  (0, 3161.96) , which each bound in that interval corresponding to  00000 and  99999 respectively.

    However, what I'm finding is not a uniform distribution at all. The distribution I am finding is the one that is attached as a histogram, with small numbers receiving a far larger share of the selection.

    Can anybody explain why this is not showing a uniform distribution, and what can be done to make it so?
    Attached Thumbnails Attached Thumbnails Why is this distribution non-uniform?-histogram.png  
    Last edited by Phugoid; March 3rd 2011 at 02:23 PM.
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  2. #2
    Member rtblue's Avatar
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    Let's take the best case scenario: a,b,c,d, and e are all 9.

    Upon substituting these numbers in..

    9.999*10^{5/2}

    and this is your maximum value.

    Now, if we take it down one notch: a,b,c,d,e are all 8. The value you get is around 900. Big step down, huh? And remember, the probability of getting all of them to be 8 is relatively low. This makes it almost inevitable that most of the values will be concentrated the the lower region of the graph.
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  3. #3
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    Could you plot a graph:
    Frequency vs lg(Number).
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  4. #4
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    Quote Originally Posted by rtblue View Post
    Let's take the best case scenario: a,b,c,d, and e are all 9.

    Upon substituting these numbers in..

    9.999*10^{5/2}

    and this is your maximum value.

    Now, if we take it down one notch: a,b,c,d,e are all 8. The value you get is around 900. Big step down, huh? And remember, the probability of getting all of them to be 8 is relatively low. This makes it almost inevitable that most of the values will be concentrated the the lower region of the graph.
    Generally, I think your explanation is a bit off, because there are others ways of generating numbers between that given by 88888 and 99999, and so the large step between the two in magnitude doesn't necessarily indicate a tendency towards smaller numbers.

    However, I've worked out why the distribution is non-uniform.

    There are just far more ways for a small number to be expressed than for a large number. For example, the only way to get a number greater than 1,000 is for the final digit to be 9, and for the other numbers to be above a certain threshold.

    By contrast, to get a number between 0 and 1, you can have a, b, c, and d be anything, and e be [0,2], or alternatively, you can have e be as high as 9, and have a, b, c, and d by low numbers, for example 00019 gives 0.316.

    So generally, it is simply that the conditions necessary for very high numbers occur with far smaller frequency than with high numbers.
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