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Math Help - Binomial distribution

  1. #1
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    Binomial distribution

    For the B(n,p) distribution, by considering \frac{p_{k}}{p_{k-1}}, show that p_{k} is largest when 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

    <br />
p_i = \frac {(n-i+1)!} {k!} p^i (1-p)^{n-i}<br />

    then

    <br />
\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)}}}<br />

    <br />
\displaystyle{=\frac {n-k+1} k \left( \frac {1-p} p \right)}<br />

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

    <br />
\displaystyle{\left( \frac {n-k+1} k \right) \left( \frac p {1-p} \right) = 1}<br />

    Solve for k.
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