Table of contents

1. Intro to Stats and Collecting Data55m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically1h 45m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables2h 33m
6. Normal Distribution and Continuous Random Variables1h 38m
7. Sampling Distributions & Confidence Intervals: Mean1h 53m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample2h 19m
10. Hypothesis Testing for Two Samples3h 22m
11. Correlation1h 6m
12. Regression1h 4m
13. Chi-Square Tests & Goodness of Fit1h 20m
14. ANOVA1h 0m

5. Binomial Distribution & Discrete Random Variables

Discrete Random Variables

Statistics

5. Binomial Distribution & Discrete Random Variables

Discrete Random Variables

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2:25 minutes

Problem 4.1.27

Textbook Question

Identifying Probability Distributions In Exercises 27 and 28, determine whether the distribution is a probability distribution. If it is not a probability distribution, explain why.

Verified step by step guidance

Step 1: Understand the requirements for a probability distribution. A probability distribution must satisfy two conditions: (1) The sum of all probabilities must equal 1, and (2) each individual probability must be between 0 and 1 inclusive.

Step 2: Verify the first condition by summing all the probabilities in the table. Add P(0), P(1), P(2), P(3), and P(4). Use the formula: \( \sum P(x) = P(0) + P(1) + P(2) + P(3) + P(4) \).

Step 3: Check the second condition by ensuring that each probability value in the table (0.30, 0.25, 0.25, 0.15, 0.05) is between 0 and 1 inclusive. If any value is outside this range, the distribution is not valid.

Step 4: If both conditions are satisfied, conclude that the given distribution is a probability distribution. If either condition fails, explain why the distribution is not valid.

Step 5: Interpret the results and provide reasoning based on the checks performed in steps 2 and 3. For example, if the sum of probabilities is not equal to 1, state the discrepancy and its impact on validity.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Probability Distribution

A probability distribution describes how probabilities are assigned to each possible value of a random variable. For a distribution to be valid, the probabilities must be non-negative and sum to one. This concept is fundamental in statistics as it provides a framework for understanding the likelihood of different outcomes.

Sum of Probabilities

In a valid probability distribution, the sum of all probabilities must equal 1. This ensures that the total likelihood of all possible outcomes is accounted for. If the sum is less than or greater than 1, the distribution cannot be considered valid, indicating an error in the assignment of probabilities.

Non-negativity of Probabilities

Probabilities must be non-negative, meaning they cannot be less than zero. This principle is essential because negative probabilities do not have a meaningful interpretation in the context of likelihood. Each probability value in a distribution must reflect a real chance of occurrence, reinforcing the need for non-negativity.

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Related Practice

Textbook Question

In Exercises 1–7, consider a grocery store that can process a total of four customers at its checkout counters each minute.

The mean increases to five arrivals per minute, but the store can still process only four per minute. Generate a list of 20 random numbers with a Poisson distribution for mu = 5 . Then create a table that shows the number of customers waiting at the end of 20 minutes.

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Textbook Question

Unusual Events In Exercise 19, would it be unusual for a household to have no HD televisions? Explain your reasoning.

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Textbook Question

Determining a Missing Probability In Exercises 25 and 26, determine the missing probability for the probability distribution.

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Textbook Question

Determining a Missing Probability In Exercises 25 and 26, determine the missing probability for the probability distribution.

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Textbook Question

"True or False? In Exercises 5–8, determine whether the statement is true or false. If it is false, rewrite it as a true statement.

The mean of the random variable of a probability distribution describes how the outcomes vary."

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Multiple Choice

A student is analyzing different types of variables in a statistics class. Which of the following below is a discrete random variable?

A factory produces lightbulbs in batches of 50. The probability distribution for the number of defective lightbulbs in a randomly selected batch is shown below. Find the expected value.

A company tracks the number of complaints they receive, where the random variable X is the number of complaints received daily. Find the variance & standard deviation of this distribution.

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