1. Discrete Probability Distributions Chapter 4

2. § 4.1 Probability Distributions

3. Larson & Farber, Elementary Statistics: Picturing the World, 3e 3 Random Variables A random variable x represents a numerical value associated with each outcome of a probability distribution. A random variable is discrete if it has a finite or countable number of possible outcomes that can be listed. x 2106048 A random variable is continuous if it has an uncountable number or possible outcomes, represented by the intervals on a number line. x 2106048

4. Larson & Farber, Elementary Statistics: Picturing the World, 3e 4 Random Variables Example : Decide if the random variable x is discrete or continuous. a.) The distance your car travels on a tank of gas b.) The number of students in a statistics class The distance your car travels is a continuous random variable because it is a measurement that cannot be counted. (All measurements are continuous random variables.) The number of students is a discrete random variable because it can be counted.

5. Larson & Farber, Elementary Statistics: Picturing the World, 3e 5 Discrete Probability Distributions A discrete probability distribution lists each possible value the random variable can assume, together with its probability. A probability distribution must satisfy the following conditions. In WordsIn Symbols 1. The probability of each value of the discrete random variable is between 0 and 1, inclusive. 0  P (x)  1 2. The sum of all the probabilities is 1. ΣP (x) = 1

6. Larson & Farber, Elementary Statistics: Picturing the World, 3e 6 Constructing a Discrete Probability Distribution Guidelines Let x be a discrete random variable with possible outcomes x 1, x 2, …, x n. 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.

7. Larson & Farber, Elementary Statistics: Picturing the World, 3e 7 Constructing a Discrete Probability Distribution Example : The spinner below is divided into two sections. The probability of landing on the 1 is 0.25. The probability of landing on the 2 is 0.75. Let x be the number the spinner lands on. Construct a probability distribution for the random variable x. 2 1 x P (x) 10.25 20.75 Each probability is between 0 and 1. The sum of the probabilities is 1.

8. Larson & Farber, Elementary Statistics: Picturing the World, 3e 8 Constructing a Discrete Probability Distribution Example : The spinner below is spun two times. The probability of landing on the 1 is 0.25. The probability of landing on the 2 is 0.75. Let x be the sum of the two spins. Construct a probability distribution for the random variable x. 2 1 Continued. The possible sums are 2, 3, and 4. P (sum of 2) = 0.25  0.25 = 0.0625 “and” Spin a 1 on the first spin. Spin a 1 on the second spin.

9. Larson & Farber, Elementary Statistics: Picturing the World, 3e 9 Constructing a Discrete Probability Distribution Example continued : 2 1 P (sum of 3) = 0.25  0.75 = 0.1875 “and” Spin a 1 on the first spin. Spin a 2 on the second spin. “or” P (sum of 3) = 0.75  0.25 = 0.1875 “and” Spin a 2 on the first spin. Spin a 1 on the second spin. Sum of spins, x P (x) 20.0625 3 4 Continued. 0.375 0.1875 + 0.1875

10. Larson & Farber, Elementary Statistics: Picturing the World, 3e 10 Constructing a Discrete Probability Distribution Example continued : 2 1 P (sum of 4) = 0.75  0.75 = 0.5625 “and” Spin a 2 on the first spin. Spin a 2 on the second spin. Sum of spins, x P (x) 20.0625 30.375 4 0.5625 Each probability is between 0 and 1, and the sum of the probabilities is 1.

11. Larson & Farber, Elementary Statistics: Picturing the World, 3e 11 Graphing a Discrete Probability Distribution Example : Graph the following probability distribution using a histogram. Sum of spins, x P (x) 20.0625 30.375 40.5625 Sum of Two Spins 0 0.4 0.3 0.2 x Probability 0.6 0.1 0.5 2 34 Sum P(x)P(x)

12. Larson & Farber, Elementary Statistics: Picturing the World, 3e 12 Mean The mean of a discrete random variable is given by μ = ΣxP(x). Each value of x is multiplied by its corresponding probability and the products are added. x P (x) 20.0625 30.375 40.5625 Example : Find the mean of the probability distribution for the sum of the two spins. xP (x) 2(0.0625) = 0.125 3(0.375) = 1.125 4(0.5625) = 2.25 ΣxP(x) = 3.5 The mean for the two spins is 3.5.

13. Larson & Farber, Elementary Statistics: Picturing the World, 3e 13 Variance The variance of a discrete random variable is given by  2 = Σ(x – μ) 2 P (x). x P (x) 20.0625 30.375 40.5625 Example : Find the variance of the probability distribution for the sum of the two spins. The mean is 3.5. x – μ –1.5 –0.5 0.5 (x – μ) 2 2.25 0.25 P (x)(x – μ) 2  0.141  0.094  0.141 ΣP(x)(x – 3.5) 2 The variance for the two spins is approximately 0.376  0.376

14. Larson & Farber, Elementary Statistics: Picturing the World, 3e 14 Standard Deviation x P (x) 20.0625 30.375 40.5625 The standard deviation of a discrete random variable is given by Example : Find the standard deviation of the probability distribution for the sum of the two spins. The variance is 0.376. x – μ –1.5 –0.5 0.5 (x – μ) 2 2.25 0.25 P (x)(x – μ) 2 0.141 0.094 0.141 Most of the sums differ from the mean by no more than 0.6 points.

15. Larson & Farber, Elementary Statistics: Picturing the World, 3e 15 Expected Value The expected value of a discrete random variable is equal to the mean of the random variable. Expected Value = E(x) = μ = ΣxP(x). Example : At a raffle, 500 tickets are sold for $1 each for two prizes of $100 and $50. What is the expected value of your gain? Your gain for the $100 prize is $100 – $1 = $99. Your gain for the $50 prize is $50 – $1 = $49. Write a probability distribution for the possible gains (or outcomes). Continued.

16. Larson & Farber, Elementary Statistics: Picturing the World, 3e 16 Expected Value Gain, x P (x) Example continued : At a raffle, 500 tickets are sold for $1 each for two prizes of $100 and $50. What is the expected value of your gain? Because the expected value is negative, you can expect to lose $0.70 for each ticket you buy. Winning no prize $99 $49 –$1 E(x) = ΣxP(x).

17. Larson & Farber, Elementary Statistics: Picturing the World, 3e 17 Page 179-183, 5-6, 12-24 even; 27- 31 odd, 33-35 odd 5 True 6 False, it is equal to the mean 12 Continuous 14 Discrete 16 Continuous 18 Discrete 20a 0.45 b 0.80 22 0.15

18. Larson & Farber, Elementary Statistics: Picturing the World, 3e 18 Page 179-183, 5-6, 12-24 even; 27- 31 odd, 33-35 odd 24 No the sum is not equal to 1 26 Yes

19. Larson & Farber, Elementary Statistics: Picturing the World, 3e 19 27 XP(X) 00.686 10.195 20.077 30.022 40.013 50.006 Mean = 0.497

20. Larson & Farber, Elementary Statistics: Picturing the World, 3e 20 29 XP(X) 00.432 10.403 20.137 30.029 Mean = 0.764

21. Larson & Farber, Elementary Statistics: Picturing the World, 3e 21 31 XP(X) 00.031 10.063 20.151 30.297 40.219 50.156 60.083 Mean = 3.410

22. Larson & Farber, Elementary Statistics: Picturing the World, 3e 22 homework 33a 5.3 35a 2.1

23. Larson & Farber, Elementary Statistics: Picturing the World, 3e 23 homework Page 179-183 28 - 40 even

24. Larson & Farber, Elementary Statistics: Picturing the World, 3e 24 Page 179-183 28 b≈0.5 c≈1.1 d≈1.0 E a household has on average 0.5 cats with a standard deviation of 1 XP(X) 00.7275 10.1308 20.0761 30.0292 40.0214 50.0150

25. Larson & Farber, Elementary Statistics: Picturing the World, 3e 25 Page 179-183 30 b≈1.9 c≈1.7 d≈1.3 E The average number of accidents per student is 1.9 with a standard deviation of 1.3 XP(X) 00.152 10.292 20.249 30.178 40.102 50.026

26. Larson & Farber, Elementary Statistics: Picturing the World, 3e 26 Page 179-183 32 b≈3.4 c≈3.4 d≈1.8 E The average number of school related extracurricular activities per student is 3.4 with a standard deviation of 1.8 XP(X) 00.059 10.122 20.163 30.178 40.213 50.128 60.084 70.053

27. Larson & Farber, Elementary Statistics: Picturing the World, 3e 27 Page 179-183, 34, 36 34 a ≈1.9 b≈2.6 c≈1.6 d≈1.9 e The average number of ATM transactions made in one day is 1.9 with a standard deviation of 1.6 36 a ≈1.7 b ≈ 1.0 c≈1.0 d≈1.7 e The average car occupancy is 1.7 with a standard deviation of 1.0

28. Larson & Farber, Elementary Statistics: Picturing the World, 3e 28 Page 179-183, 38 & 40 38 a ≈1.6 b ≈ 1.9 c≈1.4 d≈1.6 e The average car occupancy is 1.6 with a standard deviation of 1.4 40 A 0.432 B 0.568 C 0.568

29. Larson & Farber, Elementary Statistics: Picturing the World, 3e 29 Page 179-183, 38 & 40 38 a ≈1.6 b ≈ 1.9 c≈1.4 d≈1.6 e The average car occupancy is 1.6 with a standard deviation of 1.4 40 A 0.432 B 0.568 C 0.568

30. Larson & Farber, Elementary Statistics: Picturing the World, 3e 30 Page 179-183, 34 34 a ≈1.9 b≈2.6 c≈1.6 d≈1.9 e The average number of ATM transactions made in one day is 1.9 with a standard deviation of 1.6

31. Larson & Farber, Elementary Statistics: Picturing the World, 3e 31 Homework Page 179-183, 27, 29, 37-43 odd, 44

32. Larson & Farber, Elementary Statistics: Picturing the World, 3e 32 44 XP(X) -44983/5000=0.9966 21 (gift certificate)15/5000=0.003 446 (tent)1/5000=0.0002 3146 (boat)1/5000=0.0002 The expected value of the raffle for the person buying the ticket is -3.205

33. Larson & Farber, Elementary Statistics: Picturing the World, 3e 33 Homework Page 179-180 1-4, 7-10 and 11-21 odd And read section 4.1

34. Larson & Farber, Elementary Statistics: Picturing the World, 3e 34 Homework Page 179-183, 25, 26, 31, 42

35. Larson & Farber, Elementary Statistics: Picturing the World, 3e 35 Homework - worksheet 1 μ ≈ 1.5 σ 2 ≈ 0.75 σ ≈ 0.87 2 μ ≈ 0.60 σ 2 ≈ 0.51 σ ≈ 0.72 3 μ ≈ 2 σ 2 ≈ 1 σ ≈ 1 4 Not a probability distribution sum of probabilities not equal to 1 5 μ ≈ 0.80 σ 2 ≈ 0.48 σ ≈ 0.69

36. Larson & Farber, Elementary Statistics: Picturing the World, 3e 36 Homework Page 179-183, 5-6, 12-24 even; 27-31 odd parts a and b only, 33-35 odd part a only.