1. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 1 Probability Distributions Chapter 4 M A R I O F. T R I O L A Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman

2. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 2 Overview This chapter will deal with the construction of probability distributions by presenting possible outcomes along with relative frequencies we expect.

3. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 3 Chapter 4 Probability Distributions 4-1* Overview 4-2 Random Variables 4-3 & 4-4 Binomial Experiments 4-5* The Poisson Distribution

4. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 4 4-2 Random Variables

5. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 5 Example: 10 balls marked 0 to 9 and placed in a box. Pick one ball out from the box. Q: How to represent the outcome (i.e., the number on that ball)? Solution: Use a variable, say x, to represent the outcome ----- x is called a random variable Two meanings: (1) x is one of the 10 possible outcomes: 0,1, …, 9 (2) Each can happen with a positive chance

6. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 6 Definitions  Random Variable a variable (usually x ) that has a single numerical value (determined by chance) for each outcome of an experiment Discrete random variables have a finite number or countable number of values. Continuous random variables have infinitely many values which can be associated with measurements on a continuous scale with no gaps or interruptions.

7. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 7 Definitions  Probability Distribution gives the probability for each value of the random variable

8. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 8 Probability Distribution for Number of USAir Crashes Among Seven 0123456701234567 0.210 0.367 0.275 0.115 0.029 0.004 0+ x P(x)P(x) Table 4-1

9. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 9 Probability Histogram 0.40 0.30 0.20 0.10 0 0 1 2 3 4 5 6 7 Probability Histogram Number of USAir Crashes Among Seven Figure 4-3 Number of USAir Crashes Among Seven Probability

10. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 10 Requirements for Probability Distribution

11. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 11 Requirements for Probability Distribution P ( x ) = 1 where x assumes all possible values 

12. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 12 Requirements for Probability Distribution P ( x ) = 1 where x assumes all possible values 0  P ( x )  1 for every value of x 

13. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 13 Formula 4-1 Mean: µ =  x P (x) Formula 4-2 Variance:  2 =  [ (x – µ) 2 P(x )] Formula 4-3  2 = [  x 2 P (x )] – µ 2 (shortcut) Mean, Variance and Standard Deviation of a Probability Distribution

14. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 14 Mean, Variance and Standard Deviation of a Probability Distribution Formula 4-1 Mean: µ =  x P (x) Formula 4-2 Variance:  2 =  [ (x – µ) 2 P(x )]  2 = [  x 2 P (x )] – µ 2 (shortcut) Formula 4-4 SD:  = [  x 2 P (x )] –µ 2

15. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 15 Round off Rule for µ,  2, and  Round results by carrying one more decimal place than the number of decimal places used for the random variable x. If the values of x are integers, round µ,   2, and  to one decimal place.

16. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 16 Definition Expected Value The average value of outcomes E =  [ x P( x )]