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

*Binomial distribution is a sequence of identical of Bernoulli events

*The binomial distribution model allows us to compute the probability of observing a specified number of “successes” when the process is repeated a specific number of times (e.g., in a set of patients) and the outcome for a given patient is either a success or a failure.

• 1: The number of observations n is fixed. 2: Each observation is independent. 3: Each observation represents one of two outcomes (“success” or “failure”). 4: The probability of “success” p is the same for each outcome.

• A binomial distribution is very different from a normal distribution, and yet if the sample size is large enough, the shapes will be quite similar.

The key difference is that a binomial distribution is discrete, not continuous. In other words, it is NOT possible to find a data value between any two data values.

• Since only one out of five possible answers is correct, the probability of answering a question correctly by random is 1/5=0.2. We can find the probability of having exactly 4 correct answers by random attempts as follows.

 dbinom(4, size=12, prob=0.2) 

## [1] 0.1328756

#To find the probability of having four or less correct answers by random attempts, we apply the function dbinom with x = 0,…,4.

 dbinom(0, size=12, prob=0.2) + 
+ dbinom(1, size=12, prob=0.2) + 
+ dbinom(2, size=12, prob=0.2) + 
+ dbinom(3, size=12, prob=0.2) + 
+ dbinom(4, size=12, prob=0.2) 

## [1] 0.9274445

#Alternatively, we can use the cumulative probability function for binomial distribution pbinom.

pbinom(4, size=12, prob=0.2) 

## [1] 0.9274445

The probability of four or less questions answered correctly by random in a twelve question multiple choice quiz is 92.7%.

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

• Poisson distribution is a statistical distribution that shows how many times an event is likely to occur within a specified period of time. It is used for independent events which occur at a constant rate within a given interval of time.

• If your question has an average probability of an event happening per unit (i.e. per unit of time, cycle, event) and you want to find probability of a certain number of events happening in a period of time (or number of events)

• The difference between Binomial and Poisson Distribution is that the Binomial distribution is only for a certain frame or a probability of success and the Poisson distribution is used for events that could occur a very large number of times.

• • Here, average number of cars crossing a bridge per minute is μ = 20.

ppois(q, u, lower.tail = TRUE) is an R function that gives the probability that a random variable will be lower than or equal to a value.

We have to find the probability of having seventeen or more cars, so we will use lower.trail = FALSE and set q at 12:

ppois(12,20,lower.tail = FALSE)

## [1] 0.960988

To get a percentage, we simply need to multiply this output by 100. Now we have the answer to our question: there is a % probability of having 13 or more cars crossing the bridge in any particular minute.

1-ppois(0.001,1000)

## [1] 1

ppois(0.001, lambda = 1000, lower.tail=FALSE)

## [1] 1

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

• When random variable X has a value (continuous) with the same probability
• Discrete because Random variable X has a value of x1,x2,x3,… xk with equal probability,
• A uniform distribution is a type of probabilities where all outcomes are equally likely; each variable has the same probability to the outcome.
• Applying the runif function in R

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

*is a distribution that is widely used in statistics, especially stochastic processes

*widely used as a model in engineering and science

*Process in which events occur continuously and independently at a constant average rate.

*Exponential distribution is continuous because value x is a hose whose value is greater than or equal to zero (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

• The normal distribution is defined by the following probability density function. Standar normal distribution is distribution with mean = 0 and standar deviation = 1

• Because has an x-axis range from minus to positive

• Find value z with value x minus mean Divided standar deviation.

• 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 or a genta.

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

• Chi square distribution used for goodness of fit an observation distribution with theoretical distribution, free data analysis classification criteria, and the presumption of trust hoses for normal distributed population deviations from sample standard deviations.

• V=X^2 1 + X^2 2 + ⋯ + X^2m∼χ2m∼

• The number of normal random mods that are mutually free.

• Chi square distribution continuous because variabel X 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

• t student distribution used to test the two-way hypothesis

• because the standard deviation of the population(s) is unknown so the value is replaced by the sample standard deviation ( S )

• Assume further that Z and V are independent, then the following quantity follows a Student t distribution with m degrees of freedom.

• Normal distribution and 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

• F distribution is two free khi-squared random variables, each divided by their degree of freedom.

• F distribution is continuous distribution

• F distribution is often used in statistical testing, including variance analysis and regression analysis.

# x1 and x2 is a population standard deviation of Women and Men
#y1 and y2 is a sample standard deviation of Women and Men
#Women's data
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

#Men's data 
F2 = q/p
F2

## [1] 0.595102