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Thread: binomial distributions and limit theorems!!!

  1. #1
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    binomial distributions and limit theorems!!!

    find th smallest value of n in a binomial distribution for which we can assert:
    P{ |(Xn/n) p| < .1} ≥ .9
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  2. #2
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    Quote Originally Posted by lauren2988 View Post
    find th smallest value of n in a binomial distribution for which we can assert:
    P{ |(Xn/n) – p| < .1} ≥ .9
    Well this translates into: find the smallest $\displaystyle n$ such that:

    $\displaystyle P(p-0.1<\frac{X_n}{n} <p+0.1) \ge 0.9$

    where the rv $\displaystyle X_n$ the number of successes in $\displaystyle n$ independent trials with probability of success in a single trial of $\displaystyle p$. Now given the title of this thread I assume we are supposed to use a normal approximation to the binomial to do this, in which case we have:

    $\displaystyle \frac{X_n}{n} \sim N(p, p(1-p)/n)$.

    $\displaystyle 90\%$ of the probabilty mass of a normal RV is contained within $\displaystyle \pm 1.645$ standard deviations of the mean, so if $\displaystyle p>0.1$ we have:

    $\displaystyle 0.1 \ge 1.645 \sqrt{p(1-p)/n}$

    which will allow you to find the smallest $\displaystyle n$ for which this is true.

    Now you can do the case where $\displaystyle p<0.1$ yourself.

    CB
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