proportion of female birth
We are told that 241,945 girls and 251,527 boys were born in Paris from 1745 to 1770.
y = 241945
n = 241945 + 251527
posterior = function(theta) dbeta(theta,shape1 = y+1, shape2 = n- y + 1)
curve(posterior,from = 0,to = 1, main=paste("n = ",n,"; y = ",y),xlab="theta",ylab='posterior')
curve(posterior,from = 0.485,to = 0.495, main=paste("n = ",n,"; y = ",y),xlab="theta",ylab='posterior')
curve(posterior,from = 0.485,to = 0.495, main=paste("n = ",n,"; y = ",y, "; mle shown in gray"),xlab="theta",ylab='posterior')
abline(v=y/n,col='gray')