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Thread: Binomial distribution

  1. #1
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    Binomial distribution

    For the $\displaystyle B(n,p)$ distribution, by considering $\displaystyle \frac{p_{k}}{p_{k-1}}$, show that $\displaystyle p_{k}$ is largest when $\displaystyle k=[(n+1)p]$. This k is called the "mode" of the distribution.
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  2. #2
    MHF Contributor ebaines's Avatar
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    Given

    $\displaystyle
    p_i = \frac {(n-i+1)!} {k!} p^i (1-p)^{n-i}
    $

    then

    $\displaystyle
    \displaystyle{\frac {p_k} {p_{k-1}} = \frac {\frac {(n-k+1)!} {k!} p^k (1-p)^{n-k}} {\frac {(n-(k-1)+1)!} {(k-1)!} p^{k-1} (1-p)^{n-(k-1)}}}
    $

    $\displaystyle
    \displaystyle{=\frac {n-k+1} k \left( \frac {1-p} p \right)}
    $

    You want to find the point at which the value of $\displaystyle p_k$
    goes from being greater than 1 to less than 1, because that is the point at which the probability of $\displaystyle p_{k} < p_{k-1} $

    $\displaystyle
    \displaystyle{\left( \frac {n-k+1} k \right) \left( \frac p {1-p} \right) = 1}
    $

    Solve for k.
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