Modul 3 Praktikum PSD 2023/2024 Semester Ganjil

Versi file .R dari modul ini bisa diunduh: Modul 3 (REV).R

library(Rlab)

Distribusi Kontinu

Distribusi Normal

Digunakan saat \(n >= 30\)

pdf

dnorm(2.7,
      mean = 2,
      sd = 5)

[1] 0.07901035

pdf normal standar

dnorm(2.7,
      mean = 0,
      sd = 1)

[1] 0.01042093

x3 <- seq(-4, 4, length = 100)

plot(x3,
     dnorm(x3, mean = 0, sd = 1),
     type = 'o')

cdf

pnorm(1.96,
      mean = 0,
      sd = 1)

[1] 0.9750021

plot(x3,
     pnorm(x3, mean = 0, sd = 1),
     type = 'o')

inverse CDF

kalo cdf kan inputnya x outputnya probabilitas

gimana kalo mau inputnya probabilitas outputnya x? pake qnorm

Pr(X <= x) = 1.96 (gimana cara cari x nya?)

qnorm(0.975,
      mean = 0,
      sd = 1)

[1] 1.959964

bangkitkan data random

set.seed(101)
n <- 100
random_normal <- rnorm(n, mean = 0, sd = 1)
random_normal2 <- rnorm(n, mean = 10, sd = 8)

par(mfrow = c(1,2))
plot(density(random_normal), main = "Mean 0 Sd 1")

plot(density(random_normal2), main = "Mean 10 Sd 8")

par(mfrow = c(1,1))

Distribusi Sampling

CLT (Central Limit Theorem)

CENTRAL LIMIT THEOREM : Semakin besar sampel yang diambil, bisa didekatkan ke dist. normal

Sampling Distribution (Contoh: akan dilakukan sampling dari distribusi uniform)

unif <- c(1:8)

unif

[1] 1 2 3 4 5 6 7 8

Lihat sebarannya

mean(unif)

[1] 4.5

sd(unif)

[1] 2.44949

hist(unif, main = "Uniform Distribution", xlab = " ")

Ambil berbagai ukuran sampel, bandingkan

set.seed(211)

Sample size of 3, 1000 kali percobaan

sample_means <- c( )
for(i in 1:1000){
  sample_means[i] <- mean(sample(8, 3, replace = TRUE))
}
hist(sample_means, xlim = c(0,8), main = "Sample Size of 3", xlab = "Sample Means")

Sample size of 10, 1000 kali percobaan

sample_means <- c( )
for(i in 1:1000){
  sample_means[i] <- mean(sample(8, 10, replace = TRUE))
}
hist(sample_means, xlim = c(0,8), main = "Sample Size of 10", xlab = "Sample Means")

Sample size of 50

sample_means <- c( )
for(i in 1:1000){
  sample_means[i] <- mean(sample(8, 50, replace = TRUE))
}
hist(sample_means, xlim = c(0,8), main = "Sample Size of 50", xlab = "Sample Means")

Apa kesimpulannya? (Semakin besar ukuran sampel, semakin berbentuk lonceng -> dist.normal)

Penerapan CLT

transformasi random_normal2 menjadi normal standar (berlaku untuk distribusi kontinu apapun -> sampling dist.)

gunakan CLT -> scale

par(mfrow = c(1,2))
plot(density(random_normal2), main = "Sebelum")

plot(density(scale(random_normal2)), main = "Sesudah")

par(mfrow = c(1,1))

Distribusi Kontinu (lanjutan)

Distribusi T : Digunakan saat n<30 dan variansi tidak diketahui

pdf

dt(x = 0.5, df = 14)

[1] 0.3431707

cdf

pt(0.025, df = 14)

[1] 0.5097961

Pr(T<t)=0.05 (alpha) -> nyari t nya (t-table)

qt(.95, df = 20)

[1] 1.724718

generating random data

set.seed(121)
n <- 100
randomt <- rt(n, df = 20)

hist(randomt, breaks=50, xlim = c(-6, 4))

Distribusi lain

untuk distribusi lain, intinya tetap sama hanya sesuaikan parameternya saja

format:

pdf -> d+nama distribusi()

misal pdf poisson berarti dpois()
cdf poisson: ppois()
data random dari distribusi poisson rpois()

selengkapnya bisa cek di dokumentasi

?Distributions