[機統] 跟poisson、binomial有關的題目

看板Math作者 (dudu1005)時間12年前 (2013/09/30 00:22), 編輯推噓1(1推 0噓 13→)
留言14則, 4人參與, 6年前最新討論串1/1
The probability that a radioactive substance gives off m β-articles in a unit of time is Poisson with mean K. The probability that given a particle will strike acounter and be registered is p. (a) Show that the probability of registering m β-articles in the unit time is also Poisson distribution and find its mean. (b) Show that the number of β-articles registered is independent of the number of β-articles not registered in the unit time. 剛做考古題做到這題 不知道該怎麼設? 不知道該如何下手。 麻煩了，謝謝! -- ※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 114.34.217.177
→
09/30 12:19, , 1F
Zi = 1 if i'th article is registered, = 0 if not.
09/30 12:19, 1F
→
09/30 12:20, , 2F
X = Z1+Z2+...+Zm, given m. Y = m-X.
09/30 12:20, 2F
→
09/30 12:21, , 3F
Given m, X～bin(m,p), Y～bin(m,1-p), X+Y=m.
09/30 12:21, 3F
→
09/30 12:22, , 4F
P[X=x,Y=y] = P[m=x+y]P[X=x,Y=y|m=x+y]
09/30 12:22, 4F
→
09/30 12:23, , 5F
= (K^m e^{-K}/m!)C(m,x)p^x(1-p)^y
09/30 12:23, 5F
→
09/30 12:23, , 6F
對 m = 0,1,2,... 加總.
09/30 12:23, 6F
推
09/30 21:35, , 7F
http://ppt.cc/m-Xz 不確定題意是不是在問這樣
09/30 21:35, 7F
→
09/30 21:36, , 8F
如果是的話可以參考看看 只是看到普瓦松跟二項就想
09/30 21:36, 8F
→
09/30 21:36, , 9F
到這方法..
09/30 21:36, 9F
→
10/03 02:49, , 10F
修正昨天的錯誤:
10/03 02:49, 10F
→
10/03 02:50, , 11F
P[X=x,Y=y] = Σ_m P[m=x+y]P[X=x,Y=y|m=x+y]
10/03 02:50, 11F
→
10/03 02:51, , 12F
= Σ_m (K^m e^{-K}/m!)C(m,x)p^x(1-p)^y
10/03 02:51, 12F
→
01/02 15:32, 7年前 , 13F
01/02 15:32, 13F
→
07/07 11:28, 6年前 , 14F
到這方法.. http://yaxiv.com
07/07 11:28, 14F
‣ 更多 dudu1005 的文章
文章代碼(AID): #1II5FQVE (Math)