＊DSE Maths/AL M&S II: Probability Distribution

題目:
The number of people entering the entrance in a minute follows Poisson Distribution with mean 3.
a) Find the probability of exactly 4 people entering the entrance in a minute
b) Find the probability of exactly 4 people entering the entrance in two minutes
c) Find the probability of exactly 4 people entering the entrance in one of three minutes

做probability要小心個單位, 本身個Poisson係per minute, X~Po(3)
(a) part單位都係one minute, 所以可以就咁用本身個Poisson去計:
e^-3 * 3^4 / 4! = 0.1680
(b) part單位係two minutes, 所以係兩個Poisson, 同埋係2分鐘total有4個人,而冇限制每分鐘要有幾多人,即係有  0+4, 1+3, 2+2, 3+1, 4+0  咁多個case:
e^-3 * e^-3 * 3^4 / 4! *2 ( 0+4 , 4+0 ) + e^-3 * 3 * e^-3 * 3^3 / 3! *2 ( 1+3 , 3+1 ) + e^-3 * 3^2 / 2! * e^-3 * 3^2 / 2! ( 2+2 ) = 0.1339
or 用合體Poisson去計(唔知比唔比分): Y~Po(6)
e^-6 * 6^4 / 4! = 0.1339
(c) part單位係three minutes(其中one minute), 即係變左第二個Distribution -- Binomial, Z~B(3, 0.1680)
個p(0.1680)係one minute 4個人既prob,即係(a) part ans
P(Z=1) = 3C1 * 0.1680 * ( 1- 0.1680 )^2 = 0.3489

＊注意禁機個陣唔好只係禁0.1680,要用多幾個位,或者直頭用返e^-3 * 3^4 / 4!,咁先準＊

總之,要留意個單位!!Section B既probability九成九本身係一個distribution(例如單位係one minutes),之後就會問幾個distribution既情況(例如two minutes, one in three minutes, or until n minutes),咁就會變左一個Binomial or Geometric,之前計既prob好大機會會變左Binomial or Geometric既p