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 Larson/Farber 4th ed 2

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 CO 2 levels in the atmosphere. 5

6. 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. x 153024 6

7. 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 CO 2 levels in the atmosphere. 7 x 124302…

8. Example: Random Variables Decide whether the random variable x is discrete or continuous. Solution: Continuous random variable 8 1.x = The amount of water the people of Seattle drink per day (Million Gallons per day). x 132302…

9. Example: Random Variables Decide whether the random variable x is discrete or continuous. Solution: Discrete random variable 9 2.x = The number of umbrellas sold in Seattle in the Month of February (in thousands). x 130302…

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: 10 In Words In Symbols 1.The probability of each value of the discrete random variable is between 0 and 1, inclusive. 2.The sum of all the probabilities is 1. 0  P (x)  1 ΣP (x) = 1

11. Constructing a Discrete Probability Distribution 1. Make a frequency distribution for the possible outcomes. 2. Find the sum of the frequencies. 3. Find the probability of each possible outcome by dividing its frequency by the sum of the frequencies. 4. Check that each probability is between 0 and 1 and that the sum is 1. 11 Let x be a discrete random variable with possible outcomes x 1, x 2, …, x n.

12. Example: Constructing a Discrete Probability Distribution The number of house holds with cats in Ephrata Washington from a survey of 2668 house holds. Cats, xFrequency, f 01941 1349 2203 378 457 540 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. x012345 P(x)0.730.130.080.030.020.01 Discrete probability distribution: 13 Larson/Farber 4th ed

14. Solution: Constructing a Discrete Probability Distribution This is a valid discrete probability distribution since 1. Each probability is between 0 and 1, inclusive, 0 ≤ P(x) ≤ 1. 2. 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 x012345 P(x ) 0.730.130.080.030.020.01

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. xP(x) 00.73 10.13 20.08 30.03 40.02 50.01 Example: Finding the Mean The probability distribution for the cats in Ephrata Washington. Find the mean. Solution: 17 xP(x) 0*0.730 1*0.130.13 2*0.080.15 3*0.030.09 4*0.020.09 5*0.010.07

18. Variance and Standard Deviation Variance of a discrete probability distribution σ 2 = Σ(x – μ) 2 P(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) xP(x) 00.73 10.13 20.08 30.03 40.02 50.01 19 Larson/Farber 4th ed xP(x)x - μ(x - μ) 2 (x - μ) 2 P(x) 00.73 -0.530.280.21 10.13 0.470.220.03 20.08 1.472.160.16 30.03 2.476.100.18 40.02 3.4712.030.26 50.01 4.4719.970.30

20. Solution: Finding the Variance and Standard Deviation Standard Deviation: Variance: 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, subtract the 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) Gain, x$498$248$148$73–$2 P(x) 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 Larson/Farber 4th ed 25