Girl ratio

https://stats.stackexchange.com/questions/93830/expected-number-of-ratio-of-girls-vs-boys-birth

The Zorganian Republic has some very strange customs. Couples only wish to have female children as only females can inherit the family’s wealth, so if they have a male child they keep having more children until they have a girl. If they have a girl, they stop having children. What is the ratio of girls to boys in Zorgania?

Let $X$ be the number of children in a family, it is geometric random variable with $p=0.5$, i.e., $$ P(X = x) = 0.5^x, x=1,2,3… $$ which implies $E(X) = 2$

Suppose there are $N$ families in the country, the girl ratio is $$ \frac{N}{ \sum X_i} $$

Since $\sum X_i /N \rightarrow E(X) = 2$ (law of large number), the ratio coverages to 1/2 if $N \rightarrow \infty$.

If there are only finite families, let $T$ be the total number of children of the country: $T = \sum X_i$, then $T$ has a negative binomial distribution with pmf

$$ P(T=t) = C^{t-1}_{N-1} 0.5^t, t = N, N+1… $$

It implies $$ E\left[ \frac{N}{\sum X_i} \right] = E\left[ \frac{N}{T} \right] = \sum_{t=N}^{\infty} \frac{N}{t} C^{t-1}_{N-1} 0.5^t = {_2F_1} (N, 1, N+1, -1) $$ where $_2F_1$ is the hypergeometric function.

Therefore the expected girl ratio is ${_2F_1} (N, 1, N+1, -1)$.