4
$\begingroup$

Consider a set $\Omega$ with $N$ distinct members, and a function $f$ defined on $\Omega$ that takes the values 0,1 such that $ \frac{1}{N} \sum_{x \in \Omega } f(x)=p$. For a subset $S⊆Ω$ of size n, define the sample proportion $p:= p(S)= \frac{1}{n} \sum_{x\in S} f(x)$. If each subset of size $n$ is chosen with equal probability, calculate the expectation and standard deviation of the random variable $p$.

  • 0
    For each subset containing $x$, there is a corresponding subset without $x$, and vice versa. Therefore each $x$ appears in exactly half of all subsets. Hence...2012-06-12
  • 0
    This is a question from some old interview paper I am attempting. A more detailed solution would be helpfull.2012-06-12
  • 0
    It's fairly easy to prove that the expectation is p, but for the standard deviation I have no idea. I think there is an easy way to compute this but I can't find it2012-06-12

2 Answers 2