Probability Distributions (Week-2)

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 the simply probability that has two possible outcomes, success or failure outcome in an experiment or survey that is repeated multiple times. Binomial events are a sequence of identical bernoulli events.

• Binomial distribution is used for getting likelihood a given outcome a precise number of time.

• The requirements needed are :
  • The experiment consists of n identical trials,
  • Each trial results in one of the two outcomes, called success and failure,
  • The probability of success, denoted p, remains the same from trial to trial, and
  • The n trials are independent. That is, the outcome of any trial does not affect the outcome of the others.

• No, isn’t a normal distribution. Binomial distribution is discrete and normal distribution is continuous.

• Because there is only one out of five possible answers is correct, the probability of answering a question correctly by random is 1/5=0.2.

dbinom(4, size = 20, prob = 0.2)

## [1] 0.2181994

To find the probability, use the cumulative probability function for binomial distribution pbinom.

pbinom(4, size = 20, prob = 0.2)

## [1] 0.6296483

So, the probability of four or less questions answered correctly by random in a twenty question multiple choice quiz is 62.9%.

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 often used to model the frequency with which a specified event occurs during a particular period of time.

• is used to describe the distribution of rare events in a large population.

• The difference 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.

• The answer :

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

## [1] 0.9338724

So, if there are twenty cars crossing a bridge per minute on average, the probability of having thirteen or more cars crossing the bridge in a particular minute is 96.1%.

• The answer :

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

## [1] 1

So, the probability is 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 all outcomes have equal probability.

• Uniform distribution is continuous.

• That is, all the random variables have the same probability value.

• These are the function, dunif gives the density, punif gives the distribution function qunif gives the quantile function and runif generates random deviates. example : dunif(x, min, max) and punif(x, min, max).

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

• The example is number of hours until next arrivals.

• It’s used to model the time elapsed between events.

• Meanly, events must occur at a constant rate.

• Exponential distribution is continuous.

• The checkout processing rate is equals to one divided by the mean checkout completion time. So, the processing rate is 1/3 checkouts per minute.

pexp(2, rate = 1/3)

## [1] 0.4865829

So, the probability of finishing a checkout in less than two minutes by the cashier is 48.6%.

5 Normal Distribution

5.1 Question

• What is a normal distribution and Standard Normal Distribution 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

• Normal distribution is defined using a capital letter N, followed by the mean and variance of the distribution. Standard Normal distribution is defined a special kind of transformation when the expect value is 0 and the variance equal to 1.

• Because it’s often called the “bell-shaped” that the symmetry is equal each other.

• By find expected value and the variance.

• Normal distributions are symmetric, unimodal, asymptotic and the mean, median, and mode are all equal.

• The answer :
  • when percent of people earn less than $40,000

pnorm(40000, mean = 50000, sd = 20000, lower.tail = T)

## [1] 0.3085375

• when percent of people earn between $45,000 and $65,000

pnorm(65000, mean = 50000, sd = 20000, lower.tail = T) - pnorm(45000, mean = 50000, sd = 20000, lower.tail = T)

## [1] 0.372079

• when percent of people earn more than $70,000

pnorm(70000, mean = 50000, sd = 20000, lower.tail = F)

## [1] 0.1586553

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’s used for statistical analysis such as hypothesis testing and computing confidence intervals. beside that, it can use for goodness of fit of categorical values.

• Chi-square formula is \(X^2=Σ((O-E)^2/E)\)

• Chi distribution is a continuous probability distribution and distribution of the sum of squared standard normal deviates.

• Chi square distribution 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

• It’s used for estimate population parameters when the sample size is small and/or when the population variance is unknown.

• Because the t-test here uses one sample.

• By find degrees of freedom.

• T-distribution is similar with normal 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 a probability density function that is used especially in analysis of variance and is a function of the ratio of two independent random variables each of which has a chi-square distribution and is divided by its number of degrees of freedom.

• F distribution is a probability distribution of the F Statistic and it’s continuous distribution.

• It’s used for statistical testing.

• The F distribution is derived from the Student’s t-distribution.

• The answer : Women’s data

x1 = 30
x2 = 50
y1 = 35
y2 = 45
F1 = (y1^2/x1^2)/(y2^2/x2^2)
F1

## [1] 1.680384

Men’s data

F2 = (y2^2/x2^2)/(y1^2/x1^2)
F2

## [1] 0.595102