UNIT06B：離散機率的應用案例

💡 學習重點：離散機率的定義
  ■ Binomial[n, p]: 重複機率為p的實驗n次，其成功次數的分佈
  ■ Geomtric[p]: 重複機率為p的實驗，第一次成功前失敗次數的分佈
  ■ NBinomial[n, p]: 重複機率為p的實驗，第n次成功前失敗次數的分佈
  ■ Poisson[\(\lambda\)]: 期望值為\(\lambda\)的小機率事件發生次數的分佈

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pacman::p_load(dplyr, vcd)

【1】Death by Horse Kick

資料：Load built-in dataset into HK

par(mfrow=c(1,1), cex=0.7)
(HK = HorseKicks)

nDeaths
  0   1   2   3   4 
109  65  22   3   1 

檢定：Does it comply to Poisson Distribution?

fit = goodfit(HK, type = "poisson")
summary(fit)

     Goodness-of-fit test for poisson distribution

                     X^2 df P(> X^2)
Likelihood Ratio 0.86822  3  0.83309

係數：What is the \(\lambda\)?

fit$par

$lambda
[1] 0.61

應用：What is the probability of nDeath >= 2?

1 - ppois(1, fit$par$lambda)

[1] 0.12521

🧙 問題討論：
如果保險公司想要為國防部設計一個被馬踢死的保險：
  ■ 如果你只要只靠HorseKick這一份數據，每一軍團每年被馬踢死的次數超過5次的機率是多？
  ■ 如果我們將數據fit到理論分布上面，根據理論分佈，被馬踢死的次數超過5次的機率是多？
  ■ 以上哪一種做法才是比較合理的做法的？

What is the probability of nDeath >= 5?

1 - ppois(4, fit$par$lambda)

[1] 0.00042497

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【2】“May” in Federalist Papers

資料：Load data into Fed

(Fed = Federalist)

nMay
  0   1   2   3   4   5   6 
156  63  29   8   4   1   1 

檢定：Does it comply to Poisson Distribution?

fit <- goodfit(Fed, type = "poisson")
summary(fit)

     Goodness-of-fit test for poisson distribution

                    X^2 df   P(> X^2)
Likelihood Ratio 25.243  5 0.00012505

Does it comply to Negtive Binomial Distribution?

fit = goodfit(Fed, type = "nbinomial")
summary(fit)

     Goodness-of-fit test for nbinomial distribution

                   X^2 df P(> X^2)
Likelihood Ratio 1.964  4  0.74238

係數：What are the parameters?

fit$par

$size
[1] 1.1863

$prob
[1] 0.64376

分佈：How does the distribution looks like?

par=c(margin=c(3,3,3,1),cex=0.7)
dnbinom(0:10, fit$par$size, fit$par$prob) %>% barplot(names=0:10)

機率：How is the probability that 2 <= nMay <= 6?

pnbinom(6, fit$par$size, fit$par$prob) - pnbinom(1, fit$par$size, fit$par$prob)

[1] 0.15526

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💡 學習重點：離散機率的應用步驟
  1. 檢定分佈的種類
  2. 估計分佈的參數
  3. 推論事件的機率