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Math Help - Runs

  1. #1
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    Runs

    In an n-string,that is, a string consisting of n digits, a run is defined to be an occurrence of an integer in a string such that if the next digit contains the same integer it is the same run, and if it contains another integer we have another run. And k denotes the number of integers we can use to form a string. For instance, if we form a string of 10 digits by using the numbers 1,2,3,4,5, we have n=10 & k=5. And the number of runs in the string 2344155521 is 7 (2,3,4,1,5,2,1). The question is: what is the expected number of runs in an n-string with k possible integers?

    I tried to apply the problem to the sequence of tossing an unbiased coin noting whether it is head or tail. Then we have k=2 (h or t). Since it is a nonbiased coin, p=q=1/2 (where p and q denote the probability of having head or tail) , and then expected number of runs is

    1 + 2(n-1)*p*q = 1 + 2(n-1)*(1/2)*(1/2) = 1 + (n-1)/2. But when k is larger than 2, i don't know what to do? Can you help me?
    Last edited by enrique; March 17th 2008 at 09:55 PM.
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  2. #2
    Member SengNee's Avatar
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    Quote Originally Posted by enrique View Post
    In an n-string,that is, a string consisting of n digits, a run is defined to be an occurrence of an integer in a string such that if the next digit contains the same integer it is the same run, and if it contains another integer we have another run. And k denotes the number of integers we can use to form a string. For instance, if we form a string of 10 digits by using the numbers 1,2,3,4,5, we have n=10 & k=5. And the number of runs in the string 2344155521 is 7 (2,3,4,1,5,2,1). The question is: what is the expected number of runs in an n-digit string with k possible integers?

    I tried to apply the problem to the sequence of tossing an unbiased coin noting whether it is head or tail. Then we have k=2 (h or t). Since it is a nonbiased coin, p=q=1/2 (where p and q denote the probability of having head or tail) , and then expected number of runs is

    1 + 2(n-1)*p*q = 1 + 2(n-1)*(1/2)*(1/2) = 1 + (n-1)/2. But when k is larger than 2, i don't know what to do? Can you help me?

    The answer is a range?
    If yes, I think the answer is {x:x∈R,1≤x≤n}.
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  3. #3
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    No, the answer is an integer that should depend on both n and k.
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  4. #4
    Member SengNee's Avatar
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    Quote Originally Posted by enrique View Post
    No, the answer is an integer that should depend on both n and k.
    What is the title of the chapter?
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  5. #5
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    Sorry, but i didn't understand what you meant.
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  6. #6
    Member SengNee's Avatar
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    Quote Originally Posted by enrique View Post
    Sorry, but i didn't understand what you meant.
    Such as function, polynomial, differentiation, statistic, trigonometry, geometry coordinate...
    Last edited by SengNee; March 18th 2008 at 09:13 PM.
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