tag:blogger.com,1999:blog-3550204829234352390.post6322816960244418499..comments2024-03-27T09:13:48.546+08:00Comments on 研發養成所 ( Bridan's Blog - 4rdp, For R&D Person ): 訓練數學感 101 ─ 男女人口比例Bridanhttp://www.blogger.com/profile/17055047757114667099noreply@blogger.comBlogger5125tag:blogger.com,1999:blog-3550204829234352390.post-72916140065515244782016-06-09T21:42:59.563+08:002016-06-09T21:42:59.563+08:00=x^2(1 + 2x + 3x^2 + ...)
=x^2 (d/dx (x + x^2 + x^...=x^2(1 + 2x + 3x^2 + ...)<br />=x^2 (d/dx (x + x^2 + x^3 + ...))<br />有隱藏積分在裡面,經過微分又還原成原式,沒想到你也會微積分,這差不多到了高中三年級的程度!Bridanhttps://www.blogger.com/profile/17055047757114667099noreply@blogger.comtag:blogger.com,1999:blog-3550204829234352390.post-15592409861386020932016-06-09T20:53:36.106+08:002016-06-09T20:53:36.106+08:00嗯,不是積分,是將f'(x)轉乘d/dx f(x)的形式嗯,不是積分,是將f'(x)轉乘d/dx f(x)的形式.https://www.blogger.com/profile/16677804023065232981noreply@blogger.comtag:blogger.com,1999:blog-3550204829234352390.post-50854940986246672072016-06-09T20:46:30.510+08:002016-06-09T20:46:30.510+08:001 + 2x + 3x^2 + ...=1/(1-x)^2 是很有名的收斂級數,之前一直苦惱證明,原...1 + 2x + 3x^2 + ...=1/(1-x)^2 是很有名的收斂級數,之前一直苦惱證明,原來可以利用積分來解決。<br /><br />還有,此題目也反映了早期重男輕女的社會現象。.https://www.blogger.com/profile/16677804023065232981noreply@blogger.comtag:blogger.com,1999:blog-3550204829234352390.post-34978530735891384902016-06-09T17:26:23.785+08:002016-06-09T17:26:23.785+08:00高手的證明,確實男女比例還是 1:1。生成函數對初中生來講太深了,用機率來證明會比較易懂。高手的證明,確實男女比例還是 1:1。生成函數對初中生來講太深了,用機率來證明會比較易懂。Bridanhttps://www.blogger.com/profile/17055047757114667099noreply@blogger.comtag:blogger.com,1999:blog-3550204829234352390.post-54699161251871539722016-06-09T13:28:58.696+08:002016-06-09T13:28:58.696+08:00很有趣的題目!
假設所有家庭都會不斷生小孩,直至生出男嬰為止,違反直覺的是:每個家庭的男孩和女孩數量...很有趣的題目!<br />假設所有家庭都會不斷生小孩,直至生出男嬰為止,違反直覺的是:每個家庭的男孩和女孩數量的期望值都是1個!亦即未來男女人口比例依然接近1:1。<br />證明:<br />男孩數量期望值=1/2 x 1 + 1/4 x 1 + 1/8 x 1 + ... = 1 (幾何級數)<br />女孩數量期望值=1/2 x 0 + 1/4 x 1 + 1/8 x 2 + ... = ?<br />這裡我使用生成函數的方法計算這數列的和:<br />Let f(x)=x^2 + 2x^3 + 3x^4 + ...<br />f(1/2)就是要找的答案。<br />f(x)<br />=x^2 + 2x^3 + 3x^4 + ...<br />=x^2(1 + 2x + 3x^2 + ...)<br />=x^2 (d/dx (x + x^2 + x^3 + ...))<br />=x^2 (d/dx (x/(1-x)))<br />=x^2/(1-x)^2<br />代入x=1/2,答案正正是1!z423x5c6https://www.blogger.com/profile/07092469103664178667noreply@blogger.com