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Math Help - nearest integer function

  1. #1
    Super Member Random Variable's Avatar
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    nearest integer function

    Let x be a random real number from the interval (0, 1].

    Let nint(x) be the nearest integer function of x.

    What is the probability that nint(1/x) is even?


    Nearest integer function - Wikipedia, the free encyclopedia


    Moderator edit: Approved Challenge question.
    Last edited by mr fantastic; June 10th 2010 at 09:53 PM. Reason: Approval
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    I will be the first fool to state that I think it should be 1/2, which I suppose is wrong, otherwise I don't see why you'd pose the question. So I'm curious.

    However, I've seen "nearest integer" defined in different ways. By your definition, does nint(1/2)=0 or 1?
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  3. #3
    Moo
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    As he gave the wikipedia link, I guess we should take 0.

    But the probability of getting n/2 is 0, so it's not really important actually.
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    Quote Originally Posted by Random Variable View Post
    Let x be a random real number from the interval (0, 1].

    Let nint(x) be the nearest integer function of x.

    What is the probability that nint(1/x) is even?


    Nearest integer function - Wikipedia, the free encyclopedia
    my guess would be:

    Spoiler:
    2-\frac{\pi}{2}.
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  5. #5
    Super Member Random Variable's Avatar
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    Quote Originally Posted by AlephZero View Post
    I will be the first fool to state that I think it should be 1/2, which I suppose is wrong, otherwise I don't see why you'd pose the question. So I'm curious.

    However, I've seen "nearest integer" defined in different ways. By your definition, does nint(1/2)=0 or 1?
    Since we're dealing with a continuous distribution, I don't think it matters.
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  6. #6
    Super Member Random Variable's Avatar
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    Quote Originally Posted by NonCommAlg View Post
    my guess would be:

    Spoiler:
    2-\frac{\pi}{2}.
    Yep.
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  7. #7
    Super Member Random Variable's Avatar
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    Since NonCommAlg didn't explain how he did the problem, I'll give some hints for those still interested.


    Hint 1 :

    Spoiler:
    If  \text{nint}\big(\frac{1}{x}\big) = 2n (where  n \in  \mathbb{Z} ) , then  2n - \frac{1}{2} < \frac{1}{x} < 2n + \frac{1}{2} . (Whether or not they should be strict inequalties does not effect the calculation of the probability.)


    Hint 2 :

    Spoiler:
    Write out the terms of the series and use the fact that  1 - \frac {1}{3} + \frac{1}{5} - \frac {1}{7} + \frac {1}{9} + ... = \frac {\pi}{4}
    Last edited by Random Variable; July 10th 2009 at 04:46 PM.
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  8. #8
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by Random Variable View Post
    Since NonCommAlg didn't explain how he did the problem, I'll give some hints for those still interested.


    Hint 1 :

    Spoiler:
    If  \text{nint}\big(\frac{1}{x}\big) = 2n (where  n \in  \mathbb{Z} ) , then  2n - \frac{1}{2} < \frac{1}{x} < 2n + \frac{1}{2} . (Whether or not they should be strict inequalties does not effect the calculation of the probability.)


    Hint 2 :

    Spoiler:
    Write out the terms of the series and use the fact that  1 - \frac {1}{3} + \frac{1}{5} - \frac {1}{7} + \frac {1}{9} + ... = \frac {\pi}{4}
    Care to elaborate?
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  9. #9
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    Probably this technique is always used in nowadays MO contests .


    Consider the function f(x) = \frac{1}{x} then partition the y-axis :  1 \leq y = \frac{1}{x} < +\infty into such intervals :  [1,1.5) , [1.5,2.5) , [2.5,3.5) , [3.5,4.5) ...  then consider the intervals which are mapped to the x-axis from them :

     ( \frac{2}{3} ,1 ]  , ( \frac{2}{5} , \frac{2}{3} ] , (\frac{2}{7} , \frac{2}{5} ], (\frac{2}{9} , \frac{2}{7} ].. Shade the intervals   (\frac{2}{5} , \frac{2}{3} ] , (\frac{2}{9} , \frac{2}{7} ]..  and consider the total length of the shaded intervals which is :


     \frac{2}{3} - \frac{2}{5} + \frac{2}{7} - \frac{2}{9} + ...

     = 2 \left[ \frac{1}{3} - \frac{1}{5} + \frac{1}{7} - \frac{1}{9} + ... \right] = 2 ( 1 - \frac{\pi}{4} ) = 2 - \frac{\pi}{2}

    The required probability is the ratio of it to the unit length , also 2 - \frac{\pi}{2}

    Remark : Intuitively , if the function is replaced with step function , the answer will be something like  \ln{2}
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