# Thread: binomial distributions and limit theorems!!!

1. ## 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

2. Originally Posted by lauren2988
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 $n$ such that:

$P(p-0.1<\frac{X_n}{n}

where the rv $X_n$ the number of successes in $n$ independent trials with probability of success in a single trial of $p$. 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:

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

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

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

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

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

CB