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

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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

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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

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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

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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

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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

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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

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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

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