带可变阈值的加权生育停止规则
A Common Structure Behind Birth Stopping Rules
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中文总结 AI 辅助
该研究针对n个家庭按女孩数等于d倍男孩数加可变阈值停止生育的问题，推导了男孩期望比例与男女比例的精确级数公式，补充了相关规则的新求和结果并给出两种初等证明。
中文摘要 AI 辅助
我们研究一类停止规则问题：共n个家庭，每个家庭持续生育直至其女孩数量首次等于d倍男孩数量加上正整数阈值k_i。所有家庭停止生育后，我们求总人口中男孩的期望比例及期望男女比例，且允许男孩、女孩出生概率不等。我们得到了这两个期望的精确级数公式，该期望比例公式可还原公平首女规则与未加权多数女孩规则的已知结果。据我们所知，这类带可变阈值的一般加权规则求和是新的，我们给出了两个初等证明，分别为双射证明与概率证明。
英文摘要
We study stopping rules in which each family is assigned a positive integer and stops the first time its number of girls equals that integer plus a fixed nonnegative integer multiple of its number of boys. After all families have stopped, we ask for the expected proportion of boys and the expected boy-to-girl ratio in the combined population, and show that both expectations depend on the assigned integers only through their sum. We obtain exact series for both expectations, allowing the probabilities of a boy and a girl to be unequal. Our formulas recover the known formulas for stopping at the first girl and for stopping when girls first outnumber boys. These expectations can also be recovered from earlier general results on Galton--Watson degree profiles and shifted inverse moments, but the resulting series do not appear to have been recorded in these forms. To our knowledge, our two elementary proofs, one bijective and one probabilistic, are also new.
