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Discrete Probability Distributions: Constructing, Mean, Variance, and Expected Value

This chapter explains how to construct a discrete probability distribution, find the mean, variance, and expected value. It also distinguishes between discrete and continuous random variables.

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Discrete Probability Distributions: Constructing, Mean, Variance, and Expected Value

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  1. Chapter 4 Discrete Probability Distributions 1 Larson/Farber 4th ed

  2. Chapter Outline • 4.1 Probability Distributions • 4.2 Binomial Distributions • 4.3 More Discrete Probability Distributions 2 Larson/Farber 4th ed

  3. Section 4.1 Probability Distributions 3 Larson/Farber 4th ed

  4. Section 4.1 Objectives • Distinguish between discrete random variables and continuous random variables • Construct a discrete probability distribution and its graph • Determine if a distribution is a probability distribution • Find the mean, variance, and standard deviation of a discrete probability distribution • Find the expected value of a discrete probability distribution 4

  5. Random Variables Random Variable • Represents a numerical value associated with each outcome of a probability distribution. • Denoted by x • Examples • x = The number of single occupancy vehicles on I5 during rush hour. • x = The amount of CO2 levels in the atmosphere. 5

  6. x 0 1 2 3 4 5 Random Variables Discrete Random Variable • Has a finite or countable number of possible outcomes that can be listed. • Example • x = The number of single occupancy vehicles on I-5 during rush hour. 6

  7. x 0 1 2 3 … 24 Random Variables Continuous Random Variable • Has an uncountable number of possible outcomes, represented by an interval on the number line. • Example • x = The amount of CO2 levels in the atmosphere. 7

  8. x 0 1 2 3 … 32 Example: Random Variables Decide whether the random variable x is discrete or continuous. • x = The amount of water the people of Seattle drink per day (Million Gallons per day). Solution: Continuous random variable 8

  9. x 0 1 2 3 … 30 Example: Random Variables Decide whether the random variable x is discrete or continuous. • x = The number of umbrellas sold in Seattle in the Month of February (in thousands). Solution: Discrete random variable 9

  10. Discrete Probability Distributions Discrete probability distribution • Lists each possible value the random variable can assume, together with its probability. • Must satisfy the following conditions: In Words In Symbols 0 ≤P (x) ≤ 1 • The probability of each value of the discrete random variable is between 0 and 1, inclusive. • The sum of all the probabilities is 1. ΣP (x) = 1 10

  11. Constructing a Discrete Probability Distribution Let x be a discrete random variable with possible outcomes x1, x2, … , xn. • Make a frequency distribution for the possible outcomes. • Find the sum of the frequencies. • Find the probability of each possible outcome by dividing its frequency by the sum of the frequencies. • Check that each probability is between 0 and 1 and that the sum is 1. 11

  12. Example: Constructing a Discrete Probability Distribution The number of house holds with cats in Ephrata Washington from a survey of 2668 house holds. 12 Larson/Farber 4th ed

  13. Solution: Constructing a Discrete Probability Distribution • Divide the frequency of each score by the total number of individuals in the study to find the probability for each value of the random variable. • Discrete probability distribution: 13 Larson/Farber 4th ed

  14. Solution: Constructing a Discrete Probability Distribution This is a valid discrete probability distribution since • Each probability is between 0 and 1, inclusive,0 ≤ P(x) ≤ 1. • The sum of the probabilities equals 1, ΣP(x) = 0.73 + 0.13 + 0.08 + 0.03 + 0.02 + 0.01 = 1. 14 Larson/Farber 4th ed

  15. Solution: Constructing a Discrete Probability Distribution • Histogram Because the width of each bar is one, the area of each bar is equal to the probability of a particular outcome. 15 Larson/Farber 4th ed

  16. Mean Mean of a discrete probability distribution • μ = ΣxP(x) • Each value of x is multiplied by its corresponding probability and the products are added. 16 Larson/Farber 4th ed

  17. Example: Finding the Mean The probability distribution for the cats in Ephrata Washington. Find the mean. Solution: 17

  18. Variance and Standard Deviation Variance of a discrete probability distribution • σ2 = Σ(x – μ)2P(x) Standard deviation of a discrete probability distribution 18 Larson/Farber 4th ed

  19. Example: Finding the Variance and Standard Deviation The probability distribution for cats per household in Ephrata, Wa. Find the variance and standard deviation. ( μ = 0.53) 19 Larson/Farber 4th ed

  20. Solution: Finding the Variance and Standard Deviation Variance: Standard Deviation: 20 Larson/Farber 4th ed

  21. Expected Value Expected value of a discrete random variable • Equal to the mean of the random variable. • E(x) = μ = ΣxP(x) 21 Larson/Farber 4th ed

  22. Example: Finding an Expected Value At a raffle, 1500 tickets are sold at $2 each for four prizes of $500, $250, $150, and $75. You buy one ticket. What is the expected value of your gain? 22 Larson/Farber 4th ed

  23. Solution: Finding an Expected Value • To find the gain for each prize, subtractthe price of the ticket from the prize: • Your gain for the $500 prize is $500 – $2 = $498 • Your gain for the $250 prize is $250 – $2 = $248 • Your gain for the $150 prize is $150 – $2 = $148 • Your gain for the $75 prize is $75 – $2 = $73 • If you do not win a prize, your gain is $0 – $2 = –$2 23 Larson/Farber 4th ed

  24. Solution: Finding an Expected Value • Probability distribution for the possible gains (outcomes) You can expect to lose an average of $1.35 for each ticket you buy. 24 Larson/Farber 4th ed

  25. Section 4.1 Summary • Distinguished between discrete random variables and continuous random variables • Constructed a discrete probability distribution and its graph • Determined if a distribution is a probability distribution • Found the mean, variance, and standard deviation of a discrete probability distribution • Found the expected value of a discrete probability distribution 25 Larson/Farber 4th ed

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