1. Pendahuluan
2. Load Library
3. Simulasi Data
4. Tabel Frekuensi Observasi
5. Probabilitas Teoretik & Frekuensi Harapan
6. Grafik Observed vs Expected
7. Grafik PMF Teoretik
8. Contoh Perhitungan CDF
9. Uji Chi-Square Goodness-of-Fit
10. Estimasi λ (MLE)
11. Probabilitas Tambahan
12. Bootstrap CI
13. Kesimpulan

1. Pendahuluan

Distribusi Poisson digunakan untuk memodelkan jumlah kejadian dalam suatu interval tertentu.
Analisis ini mencakup simulasi data, statistik, visualisasi, serta uji kesesuaian distribusi.

2. Load Library

library(ggplot2)
library(fitdistrplus)

## Loading required package: MASS

## Loading required package: survival

3. Simulasi Data

set.seed(123)
lambda_true <- 3
n <- 200
x <- rpois(n, lambda_true)

mean_x <- mean(x)
var_x  <- var(x)
lambda_hat <- mean_x

cat("Jumlah observasi:", n, "\n")

## Jumlah observasi: 200

cat("Mean:", mean_x, "\n")

## Mean: 3.005

cat("Var:", var_x, "\n")

## Var: 2.487412

cat("Estimasi lambda:", lambda_hat, "\n")

## Estimasi lambda: 3.005

4. Tabel Frekuensi Observasi

tab_obs <- as.data.frame(table(x))
colnames(tab_obs) <- c("k","freq")
tab_obs$k <- as.integer(as.character(tab_obs$k))
tab_obs

##   k freq
## 1 0    6
## 2 1   25
## 3 2   55
## 4 3   44
## 5 4   36
## 6 5   20
## 7 6    9
## 8 7    4
## 9 8    1

5. Probabilitas Teoretik & Frekuensi Harapan

k_max <- max(tab_obs$k, qpois(0.999, lambda_hat))
k_vals <- 0:k_max
pmf_theo <- dpois(k_vals, lambda_hat)
expected_freq <- pmf_theo * n

obs_extended <- merge(data.frame(k=k_vals), tab_obs, by="k", all.x=TRUE)
obs_extended$freq[is.na(obs_extended$freq)] <- 0
obs_extended$expected <- expected_freq
obs_extended

##     k freq   expected
## 1   0    6  9.9077509
## 2   1   25 29.7727914
## 3   2   55 44.7336190
## 4   3   44 44.8081750
## 5   4   36 33.6621415
## 6   5   20 20.2309470
## 7   6    9 10.1323326
## 8   7    4  4.3496657
## 9   8    1  1.6338432
## 10  9    0  0.5455221
## 11 10    0  0.1639294

6. Grafik Observed vs Expected

df_plot <- obs_extended
df_plot$label <- factor(df_plot$k, levels=df_plot$k)

ggplot(df_plot, aes(x=label)) +
  geom_bar(aes(y=freq), stat="identity", alpha=0.6) +
  geom_point(aes(y=expected), size=2, color="red") +
  geom_line(aes(y=expected, group=1), color="red", linetype="dashed") +
  labs(title=paste("Observed vs Expected Poisson — λ =", round(lambda_hat,3)),
       x="Jumlah Kejadian (k)", y="Frekuensi") +
  theme_minimal()

7. Grafik PMF Teoretik

pmf_df <- data.frame(k=k_vals, pmf=pmf_theo)

ggplot(pmf_df, aes(x=k, y=pmf)) +
  geom_col() +
  labs(title=paste("PMF Distribusi Poisson (λ =", round(lambda_hat,3),")"),
       x="k", y="P(X=k)") +
  theme_minimal()

8. Contoh Perhitungan CDF

cat("P(X <= 2) =", ppois(2, lambda_hat), "\n")

## P(X <= 2) = 0.4220708

cat("P(X >= 4) =", 1 - ppois(3, lambda_hat), "\n")

## P(X >= 4) = 0.3538883

9. Uji Chi-Square Goodness-of-Fit

9.1 Penggabungan Tail

combine_tail <- function(k, obs, exp, min_expected = 5) {
  k2 <- k ; obs2 <- obs ; exp2 <- exp
  
  while(any(exp2 < min_expected) && length(exp2) > 1) {
    i <- length(exp2)
    exp2[i-1] <- exp2[i-1] + exp2[i]
    obs2[i-1] <- obs2[i-1] + obs2[i]
    k2 <- k2[-i]
    exp2 <- exp2[-i]
    obs2 <- obs2[-i]
  }
  data.frame(k=k2, obs=obs2, exp=exp2)
}

comb_df <- combine_tail(obs_extended$k, obs_extended$freq, obs_extended$expected)
comb_df

##   k obs       exp
## 1 0   6  9.907751
## 2 1  25 29.772791
## 3 2  55 44.733619
## 4 3  44 44.808175
## 5 4  36 33.662141
## 6 5  20 20.230947
## 7 6   9 10.132333
## 8 7   5  6.692960

9.2 Perhitungan Manual

chisq_stat <- sum((comb_df$obs - comb_df$exp)^2 / comb_df$exp)
df_chisq <- nrow(comb_df) - 1 - 1
p_value <- 1 - pchisq(chisq_stat, df = df_chisq)

cat("Chi-square:", chisq_stat, "\n")

## Chi-square: 5.39687

cat("df:", df_chisq, "\n")

## df: 6

cat("p-value:", p_value, "\n")

## p-value: 0.4940079

9.3 Chi-Square dengan Fungsi R

chisq_builtin <- suppressWarnings(
  chisq.test(
    x = comb_df$obs,
    p = comb_df$exp / sum(comb_df$exp),
    rescale.p = FALSE
  )
)
chisq_builtin

## 
##  Chi-squared test for given probabilities
## 
## data:  comb_df$obs
## X-squared = 5.3953, df = 7, p-value = 0.6118

10. Estimasi λ (MLE)

fit <- fitdist(x, "pois", method = "mle")
fit

## Fitting of the distribution ' pois ' by maximum likelihood 
## Parameters:
##        estimate Std. Error
## lambda    3.005  0.1225765

11. Probabilitas Tambahan

cat("P(X >= 6) =", 1 - ppois(5, lambda_hat), "\n")

## P(X >= 6) = 0.08442288

12. Bootstrap CI

B <- 2000
boot_means <- replicate(B, mean(sample(x, replace = TRUE)))
quantile(boot_means, c(0.025, 0.975))

##     2.5%    97.5% 
## 2.799875 3.230000

13. Kesimpulan

Mean dan varians mendekati λ → karakteristik Poisson terpenuhi.
Grafik observed vs expected mendukung kecocokan distribusi Poisson.
Uji Chi-square menghasilkan p-value besar → model Poisson cocok.
Estimasi λ stabil melalui MLE dan bootstrap.

Analisis Poisson selesai dilakukan.

Analisis Distribusi Poisson

Kelompok 6