Tugas Pendalaman Materi - Pertemuan 5

1. Variabel Acak (Klasifikasi)

• (a) Jumlah komentar per unggahan: Diskrit
  • Alasan: Dapat dihitung (\(0, 1, 2, \dots\)).
• (b) Durasi video yang ditonton (detik): Kontinu
  • Alasan: Nilai desimal berapa pun mungkin dalam suatu interval.

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2. Distribusi Binomial

Perhitungan Manual:

• a) \(P(X = 2)\): \[P(X=2) = \binom{10}{2} (0{,}25)^2 (0{,}75)^8 = 45 \times 0{,}0625 \times 0{,}100113 \approx 0{,}2816\]
• b) Rata-rata \(E(X)\): \[E(X) = n \cdot p = 10(0{,}25) = 2{,}5 \text{ pengguna}\]
• c) Deviasi Standar \(SD(X)\): \[\text{Var}(X) = n \cdot p \cdot (1-p) = 10(0{,}25)(0{,}75) = 1{,}875\] \[SD(X) = \sqrt{1{,}875} \approx 1{,}3693 \text{ pengguna}\]

Verifikasi RStudio:

n <- 10; p <- 0.25

# a) P(X = 2)
dbinom(2, size = n, prob = p)

## [1] 0.2815676

# b) Mean & SD
n * p

## [1] 2.5

sqrt(n * p * (1 - p))

## [1] 1.369306

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3. Distribusi Poisson

Perhitungan Manual:

• a) \(P(X = 5)\): \[P(X=5) = \frac{e^{-8} (8^5)}{5!} = \frac{0{,}00033546 \times 32768}{120} \approx 0{,}0916\]

Verifikasi RStudio:

lambda <- 8

# a) P(X = 5)
dpois(5, lambda = lambda)

## [1] 0.09160366

# b) P(X > 12) = 1 - P(X <= 12)
1 - ppois(12, lambda = lambda)

## [1] 0.0637972

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4. Standardisasi Z & Distribusi Normal

Perhitungan Manual:

• a) \(P(X > 4)\): \[Z = \frac{4 - 3}{0{,}5} = 2{,}00\] \[P(X > 4) = 1 - P(Z \le 2{,}00) = 1 - 0{,}9772 = 0{,}0228\]

• b) Batas waktu 5% loading TERLAMA: Cari \(Z\) dengan \(\Phi(Z) = 0{,}95 \rightarrow Z \approx 1{,}645\) \[X = \mu + Z \cdot \sigma = 3 + 1{,}645(0{,}5) = 3{,}8225 \text{ detik}\]

Verifikasi RStudio:

mu <- 3; sigma <- 0.5

# a) P(X > 4)
1 - pnorm(4, mean = mu, sd = sigma)

## [1] 0.02275013

# b) Batas waktu 5% terlama
qnorm(0.95, mean = mu, sd = sigma)

## [1] 3.822427