但是樓主問的是期望值,不是趨向值..
Let \(T\)
be \(\sum X_i\)
, \(T\)
has a negative binomial distribution with pmf
\[ P(T=t) = C^{t-1}_{N-1} 0.5^t, t = N, N+1... \]
\[ E\left[ \frac{N}{T} \right] = \sum_{t=N}^{\infty} \frac{N}{t} C^{t-1}_{N-1} 0.5^t = {_2F_1} (N, 1, N+1, -1) \]
where \(_2F_1\)
is the hypergeometric function.
所以N個家庭的期望值是 \( 1- {_2F_1} (N, 1, N+1, -1)\)