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Math Help - Binomial Distribution Related Proof

  1. #1
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    Binomial Distribution Related Proof

    I'm having a tough time with this problem:



    I managed to do the proof (part a) no problem. However, I don't know where to start when it comes to part b.

    If somebody could help me get started that'd be great.
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  2. #2
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    All you need to do is check if P(k+1) is bigger or less than P(k).

    This is determined by the multiplier:
    \frac{p}{1-p}\frac{n-k}{k+1}

    and you need to check when
    \frac{p}{1-p}\frac{n-k}{k+1} < 1

    because that's when the P() terms start to decrease.
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  3. #3
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    I still don't see how to do it. Lets say I want to determine if P(k=0) is less than or equal to P(k=1). This would allow me to show that the function starts by increasing. However, if you plug the values in, you get \frac{p^2(n^2-n)}{2(1-p)^2} and \frac{pn}{1-p}. I don't see how to determine which one is greater without knowing the values.

    That being sad, I also don't know how to determine when \frac{p}{1-p}\frac{n-k}{k+1} < 1 for the same reason. It's too abstract, I don't see how you can determine it without knowing some of the values.
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  4. #4
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    It's algebra:
    \frac{p}{1-p}\frac{n-k}{k+1} < 1

    multiply out the denominators:

    p(n-k) < (1+k)(1-p)

    rearrange:

    pn-pk < 1+k -p-pk
    pn+p-1 < k
    k > pn+p-1

    So the multiplier is less than 1 when k > pn+p-1. That means the largest P(k) happened before this...
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  5. #5
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    Ah it makes sense now. I understand the problem more clearly. Thanks for you help.
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