Probability Distributions

1 Binomial Distribution

1.1 Question

• What is a binomial distribution in Statistics?
• What is binomial distribution used for?
• Please argue 4 requirements needed to be a binomial distribution?
• Is a binomial distribution a normal distribution?
• Suppose there are twenty multiple choice questions in an Statistics class quiz. Each question has five possible answers, and only one of them is correct. Find the probability of having four or less correct answers if a student attempts to answer every question at random (Explain this exercise by using R).

1.2 Answer

1. distribusi binomial atau disebut juga distribusi bernoulli ,adalah suatu distribusi yang menggunakan variabel random diskrit yang terjadi dari dua kejadian yang berkomplemen
2. digunakan untuk memodelkan jumlah keberhasilan pada jumlah sampel n dari jumlah populasi N.
3. • setiap percobaan hanya memiliki 2 peristiwa , seprti ya /tidak , atau sukses - gagal ,
   • probabilitas suatu peristiwa adalah tetap , tidak berubah untuk setiap percobaan
   • percobaanya bersifat independen
   • jumlah atau banyaknya percobaan yang merupakan komponen percobaan binomal harus tertentu

dbinom(0, size=20, prob=0.2)+
  dbinom(1, size=20, prob=0.2)+
  dbinom(2, size=20, prob=0.2)+
  dbinom(3, size=20, prob=0.2)+
  dbinom(4, size=20, prob=0.2)

## [1] 0.6296483

2 Poisson Distribution

2.1 Question

• What is a Poisson distribution in Statistics?
• When and Why is Poisson distribution used?
• What is the difference between Binomial distribution and Poisson distribution?
• If twenty cars are crossing a bridge per minute on average, visualize and find the probability of having thirteen or more cars crossing the bridge in a particular minute (Explain this exercise by using R).
• Suppose the probability that a drug produces a certain side effect is \(p = 0.1%\) and \(n = 1,000\) patients in a clinical trial receive the drug. What is the probability 0 people experience the side effect by using visualization techniques? (Explain this exercise by using R)

2.2 Answer

Type your answer here! 1. adalah kasus khusus dari distribusi binomial, dimana distribusi binomial akan menjadi distribusi poisson ketika n mendekati tak hingga dan p mendekati nol. 2. untuk jumlah kejadian pada interval tertentu seperti jarak, luas, atau volume). 3. Perbedaan utama antara Distribusi Binomial dan Poisson adalah bahwa distribusi Binomial hanya untuk kerangka tertentu atau kemungkinan keberhasilan dan distribusi Poisson digunakan untuk kejadian yang dapat terjadi dalam jumlah yang sangat besar. 4.

1-ppois(13,20)

## [1] 0.9338724

ppois(13,lambda=20, lower.tail = FALSE)

## [1] 0.9338724

p=0.001
n=1000

3 Uniform Distribution (Continuous or Discrete)

3.1 Question

• How do you find the continuous uniform distribution?
• Is the uniform distribution discrete or continuous?
• What does it mean to have a uniform distribution?
• How do you solve uniform distribution problems using R?

3.2 Answer

1. Ketika variabel acak X memiliki nilai (kontinu) dengan probabilitas yang sama
2. Diskrit karena variabel acak X bernilai x1, x2, x3, … xk dengan probabilitas yang sama,
3. Distribusi seragam adalah jenis probabilitas di mana semua hasil kemungkinan besar sama; setiap variabel memiliki probabilitas yang sama terhadap hasil.

4 Exponential Distribution

4.1 Question

• What is exponential distribution example?
• What is the exponential distribution used for?
• What is exponential distribution rate?
• Is exponential distribution discrete or continuous?
• Suppose the mean checkout time of a supermarket cashier is three minutes. Find the probability of a customer checkout being completed by the cashier in less than two minutes (Explain this exercise by using R).

4.2 Answer

1. adalah satu distribusi yang banyak digunakan di statistika khususnya proses stokastik
2. banyak digunakan sebagai model di bidang teknik dan sains
3. Proses di mana peristiwa-peristiwa terjadi secara terus menerus dan independen dengan laju rata-rata yang konstan
4. kontinu karena nilai x adalah sebagai selang yang nilainya lebih besar dari atau sama dengan nol (X≥0).

pexp(2, rate=1/3)

## [1] 0.4865829

5 Normal Distribution

5.1 Question

• What is a normal distribution and Standard Normal Distrubution in Statistics?
• Why is it called normal distribution?
• How do you calculate normal distribution?
• What are the characteristics of a normal distribution?
• The annual salaries of employees in a large company are approximately normally distributed with a mean of $50,000 and a standard deviation of $20,000 (Explain this exercise by using R).
  • What percent of people earn less than $40,000?
  • What percent of people earn between $45,000 and $65,000?
  • What percent of people earn more than $70,000?

5.2 Answer

1. distribusi normal atau disebut juga sabagai gaussian distribution adalah distribusi yang paling penting di antara distribusi yang lain. kurva dari distribusi normal mempunyai bentuk setangkup seperti lonceng.

distribusi normal standar adalah distribusi peubah acak dengan rata-rata μ=0 dan varian σ2=1 Peubah acak (variabel random) distribusi normal baku dinotasikan dengan Z yang merupakan hasil transformasi dari peubah acak X yang berdistribusi normal

2. Karena sumbu x memiliki range dari minus sampai positif
3. Dengan menentukan nilai z dengan nilai x minus rata-rata dibagi dengan standar deviasi.
4. Karakteristik distribusi normal:

1. Rata-rata terletak di tengah distribusi dan distribusinya simetris di sekitar garis tegak lurus yang ditarik melalui rata-rata,
2. Luas total di bawah kurva normal adalah 1,
3. Simpangan baku σ yang merupakan penentu lebar kurva. Semakin runcing kurva jika semakin kecil σ,
4. Kurva berbentuk seperti lonceng.

pnorm(40000,mean=50000,sd=20000)

## [1] 0.3085375

pnorm(65000,mean=50000,sd=20000) -
pnorm(45000,mean=50000,sd=20000)

## [1] 0.372079

pnorm(70000,mean=50000,sd=20000)

## [1] 0.8413447

6 Chi-squared Distribution

6.1 Question

• What is the chi square distribution used for?
• What is the chi square distribution formula?
• What is the chi distribution?
• Is Chi square distribution continuous?
• 256 visual artists were surveyed to find out their zodiac sign. The results were: Aries (29), Taurus (24), Gemini (22), Cancer (19), Leo (21), Virgo (18), Libra (19), Scorpio (20), Sagittarius (23), Capricorn (18), Aquarius (20), Pisces (23). Test the hypothesis that zodiac signs are evenly distributed across visual artists (Explain this exercise by using R).

6.2 Answer

1.Uji Chi Square berguna untuk menguji hubungan atau pengaruh dua buah variabel nominal dan mengukur kuatnya hubungan antara variabel yang satu dengan variabel nominal lainnya

3.Distribusi chi square kontinu karena variabel X kontinu # Student \(t\) Distribution {.tabset .tabset-fade .tabset-pills}

6.3 Question

• What is the Student \(t\) distribution used for?
• Why is it called Student \(t\) distribution?
• How do you find the Student \(t\) distribution?
• What kind of distribution is the \(t\) distribution?
• Philips Corporation manufactures light bulbs. The CEO claims that an average Acme light bulb lasts 300 days. A researcher randomly selects 15 bulbs for testing. The sampled bulbs last an average of 290 days, with a standard deviation of 50 days. If the CEO’s claim were true, what is the probability that 15 randomly selected bulbs would have an average life of no more than 290 days? (Explain this exercise by using R)

Note: There are two ways to solve this problem, using the T Distribution Calculator. Both approaches are presented below. Solution A is the traditional approach. It requires you to compute the t statistic, based on data presented in the problem description. Then, you use the T Distribution Calculator to find the probability. Solution B is easier. You simply enter the problem data into the T Distribution Calculator. The calculator computes a t statistic “behind the scenes”, and displays the probability. Both approaches come up with exactly the same answer.

6.4 Answer

1. untuk memperkirakan interval rata rata , untuk menguji hipotesis tentang rata rata suatu sampel , menunjukkan batas penerimaan suatu hipotesis , untuk menguji suatu pernyataan apakah sudah layak untuk dipercaya
2. karena simpangan baku dari populasi tidak diketahui maka nilainya diganti dengan simpangan baku sampel (S)
3. Asumsikan bahwa Z dan V tidak saling bergantung, maka besaran berikut mengikuti distribusi t Student dengan derajat kebebasan m.
4. adalah sebagai normal distribution dan chi square distribution

7 F Distribution

7.1 Question

• What does F distribution mean?
• What does the F distribution tell you?
• What is an F distribution used for?
• How is the F distribution derived?
• Suppose you randomly select 7 women from a population of women, and 12 men from a population of men. The table below shows the standard deviation in each sample and in each population. (Compute the f statistic, Explain this exercise by using R)

Population	Population standard deviation	Sample standard deviation
Women	30	35
Men	50	45

7.2 Answer

1.merupakan distribusi probabilitas kontinu

3.sering kali digunakan dalam pengujian statistika, antara lain analisis varians dan analisis regresi.

#x1 dan x2 adalah standard deviation pupulasi dari laki laki dan perempuan
#y1 dan y2 adalah standard deviation sampel dari laki laki dan perempuan
#data perempuan
x1=30 
x2=50
y1=35
y2=45
p =(y1^2/x1^2)
q = (y2^2/x2^2)
F1 = p/q
F1

## [1] 1.680384

#data laki laki
F2 = q/p
F2

## [1] 0.595102