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1 Written Assignment Unit 4 MATH 1280 Introduction to Statistics University of the People Jessica (Instructor) Nov 12, 2023

2 Given the table below, answer the following questions: Probability distributions are a fundamental concept in statistics. They describe the likelihood of different outcomes occurring in a random event. Probability tables are a useful way to represent probability distributions. They show the probability of each possible outcome of the event. Probability tables provide a quick, easy way to calculate the probability of different outcomes in a random event. They also help to identify outliers and patterns in the data, which can be useful for making decisions about future events (Illowsky et al., 2022). Completing the Missing Probability In the table provided, the missing probability is the probability of outcome 4. To complete the table, we need to make sure that the sum of all the probabilities is equal to 1. This is because the sum of the probabilities represents the total probability of the event occurring, which must be equal to 1. This can be done by multiplying each probability by 4. For example, the probability of outcome 1 is multiplied by 4, the probability of outcome 2 is multiplied by 4 and so on. After multiplying all the probabilities by 4, the total probability is equal to 1. (Illowsky et al., 2022). Therefore, the missing probability is: 1 - (0.12 + 0.18 + 0.30 + 0.15 + 0.10 + 0.05) = 0.20 Expected Value

3 The expected value of a probability distribution is the average of the possible outcomes, weighted by their probabilities. To calculate the expected value, we multiply each outcome by its probability and then sum the products: E(X) = Σ x * P(x) where: E(X) is the expected value x is an outcome P(x) is the probability of outcome x Using the table provided, we can calculate the expected value as follows: E(X) = (0 * 0.12) + (1 * 0.18) + (2 * 0.30) + (3 * 0.15) + (4 * 0.20) + (5 * 0.10) + (6 * 0.05) = 2.37 Therefore, the expected value of the probability distribution is 2.37. Standard Deviation The standard deviation of a probability distribution is a measure of how spread out the outcomes are. It is calculated by taking the square root of the variance, which is the average of the squared deviations from the mean: σ = √Var(X)

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4 where: σ is the standard deviation Var(X) is the variance To calculate the variance, we first need to calculate the squared deviations from the mean for each outcome: (x - E(X))^2 where: x is an outcome E(X) is the expected value We then multiply each squared deviation by its probability and sum the products: Var(X) = Σ (x - E(X))^2 * P(x) Using the table provided, we can calculate the variance as follows: Var(X) = (0 - 2.37)^2 * 0.12 + (1 - 2.37)^2 * 0.18 + (2 - 2.37)^2 * 0.30 + (3 - 2.37)^2 * 0.15 + (4 - 2.37)^2 * 0.20 + (5 - 2.37)^2 * 0.10 + (6 - 2.37)^2 * 0.05 = 1.61

5 Therefore, the variance of the probability distribution is 1.61. To calculate the standard deviation, we simply take the square root of the variance: σ = √1.61 = 1.27 Therefore, the standard deviation of the probability distribution is 1.27. The missing probability is 0.20. The expected value of the probability distribution is 2.37. The standard deviation of the probability distribution is 1.27 The expected value and standard deviation can also be used to estimate the probability of different outcomes. This probability can then be used to make decisions about the best course of action to take in a given situation (Illowsky et al., 2022). N# Question Answers 1 Complete the missing probability. 0.20. 2 What is the expected value from the table? 2.37. 3 Find the standard deviation 1.27

6 In conclusion, This probability table could represent the distribution of possible outcomes of a random experiment, such as rolling a die or flipping a coin multiple times. The expected value and standard deviation can be used to describe the distribution and to make predictions about the outcomes of future experiments. References: Illowsky, B., Dean, S., Birmajer, D., Blount, B., Boyd, S., Einsohn, M., Helmreich, Kenyon, L., Lee, S., & Taub, J. (2022). Introductory statistics. Openstax, Ch. 2 Bringing It Together: Homework - Introductory Statistics | OpenStax .

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