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

• In Statistics, Binomial Distribution is also called discrete distribution. It is used to show of summarize from the number of trials and used to determine the probability.
• It is often used to predict who will win in the election, lifespan of someome.
• 1. Experiment consist of n trials.
  2. Each trial result have 2 outcomes it is success and failure.
  3. Probability of success, denoted p and it remains the same from trial to trial.
  4. The trials are independent.
• Binomial Distribution is different from normal distribution, but in some cases the shape might be similar.

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

## [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

• In Statistics, Poisson distribution is statistical distribution that shows how many times an event is likely to occur within a specified period of time, it is also in discrete function so it is measured by occuring or not occuring.
• When you want to find probability of a certain number of events happening in a period of time, and it is used to predict probability of certain events from happening when you know the history of the event (how many time the event has occured).
• Binomial distribution describes the distribution of binary data from a finite sample. Thus it gives the probability of getting r events out of n trials. Poisson distribution describes the distribution of binary data from an infinite sample.

1-ppois(13,20)

## [1] 0.9338724

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

• By having a sample like a dice, each face has same probability to apear which is 1/6.
• Uniform Distribution is continuous.
• All the probability values are equal.
• By using dunif, punif, qunif, or runif functions.

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

• Example of exponential distribution is long distances, time for an earthquake to occur.
• To process events that occur continuously and independetnly at a constant rate.
• It is continuous.

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

• The normal distribution is defined by the following probability density function, where \(\mu\) is the population mean and \(\sigma^2\) is the variance. Standard normal distribution is distribution with mean = 0 and standar deviation = 1
• Because has an x-axis range from low to higher value
• Find value \(z\) with value \((x-\mu)/sd\).
• Characteristics of a normal distribution:

1. The average is located in the middle of the distribution and the distribution is symmetrical around a straight upright line drawn through the average.
2. The total area below the normal curve is 1.
3. Standard deviation σ which is the determinant of the width of the curve. The more pointed the curve will be when the smaller the σ.
4. The curve is shaped like a bell.

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

• It is used for goodness of fit in an observation with theorotical distribution.
• \(X^2_c=\Sigma[(O_i-E_i)^2/E_i]\)
• The number of normal random mods that are mutually free.
• It is continuous.

7 Student \(t\) Distribution

7.1 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.

7.2 Answer

• For 2 way hypothesis
• Because \(sd\) from population is unknown, so replaced by sample’s \(sd\).
• Let’s assume that \(Z\) and \(V\) are independent, then the following quantity follows Student’s t distribution with \(m\) \(df\)(degree of freedom).
• Normal Distribution and \(X^2_c\) (chi square) distribution.

8 F Distribution

8.1 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

8.2 Answer

• It is 2 of free chi squared random variables divided by their \(df\).
• It is alike to normal distribution, the diferrence is it is skewed distribution, similar to chi squared.
• For testing hypotheses about the equality of 2 population variances and testing validity of a multiple regression equation.

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

F2 = q/p
F2

## [1] 0.595102