UNIT08C：Cases of Application Prob. (1)

💡 Theoretical Discrete Distributions

• Binomial[n, p]: the distribution of the number of successes, while repeating an experiment of success rate p for n times.
• Geometric[p]: the distribution of the number of failures before the first success, while repeating an experiment of success rate p.
• NBinomial[n, p]: the distribution of the number of failures before the n-th success, while repeating an experiment of success rate p.
• Poisson[\(\lambda\)]: the distribution of the number of successes given the expected value \(\lambda\).

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pacman::p_load(dplyr, vcd)

【1】Death by Horse Kick

HorseKick: Number of events per year per corps

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

nDeaths
  0   1   2   3   4 
109  65  22   3   1 

Test H0: The data fits Poisson Distribution

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

     Goodness-of-fit test for poisson distribution

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

p=0.833 > 0.05：The data is not significantly different from Poisson.

Parameters：What is the \(\lambda\)?

fit$par

$lambda
[1] 0.61

Application：What is the probability of nDeath >= 2

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

[1] 0.12521

🧙 Discussion：
If an insurance company want to design an policy for death by horse kick, and you need to know P[nDeath>5]：
  ■ Can you estimate the Probability from the data？
  ■ Can you estimate it by the model？
  ■ Which way is better？

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

Data Federalist：In an set of Federal Papers, the numbers of “may” per paragraph.

Federalist

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

Test：H0: The data fits Poisson Distribution

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

     Goodness-of-fit test for poisson distribution

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

• It significantly differs from Poisson

H0: The data fits Negative Binomial Distribution

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

     Goodness-of-fit test for nbinomial distribution

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

• It does not significantly differs from Negative Binomial

Parameters：What are the parameters？

fit$par

$size
[1] 1.1863

$prob
[1] 0.64376

• It complies to NBinom[n=1.19, p=0.64].

Plot：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)

Estimation：What is the probability that 2 <= nMay <= 6?

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💡 Steps to Apply Theoretical Distribution

1. Test for fitness
   
2. Estimate Parameters
   
3. Estimate Probabilities of Events