1. 1 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY S TATISTICS Chapter 4 Probability Distributions

2. 2 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Chapter 4 Probability Distributions 4-1 Overview 4-2 Random Variables 4-3 Binomial Probability Distributions 4-4 Mean, Variance, Standard Deviation for the Binomial Distribution 4-5 The Poisson Distribution

3. 3 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 4-1 Overview This chapter will deal with the construction of probability distributions by combining the methods of Chapter 2 with the those of Chapter 3. Probability Distributions will describe what will probably happen instead of what actually did happen.

4. 4 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Figure 4-1 Combining Descriptive Statistics Methods and Probabilities to Form a Theoretical Model of Behavior

5. 5 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 4-2 Random Variables

6. 6 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Definitions  Random Variable a variable (typically represented by x ) that has a single numerical value, determined by chance, for each outcome of a procedure  Probability Distribution a graph, table, or formula that gives the probability for each value of the random variable

7. 7 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Probability Distribution Number of Girls Among Fourteen Newborn Babies 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0.000 0.001 0.006 0.022 0.061 0.122 0.183 0.209 0.183 0.122 0.061 0.022 0.006 0.001 0.000 x P(x)P(x) Table 4-1

8. 8 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Definitions  Discrete random variable has either a finite number of values or countable number of values, where ‘countable’ refers to the fact that there might be infinitely many values, but they result from a counting process.  Continuous random variable has infinitely many values, and those values can be associated with measurements on a continuous scale with no gaps or interruptions.

9. 9 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Figure 4-3 Probability Histogram

10. 10 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Requirements for Probability Distribution P ( x ) = 1 where x assumes all possible values 

11. 11 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Requirements for Probability Distribution P ( x ) = 1 where x assumes all possible values 0  P ( x )  1 for every value of x 

12. 12 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Formula 4-1 µ =  [x P (x)] Formula 4-2  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

13. 13 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Formula 4-1 µ =  [x P (x)] Formula 4-2  2 =  [ (x - µ) 2 P (x )] Formula 4-3  2 = [  x 2 P (x )] - µ 2 (shortcut) Formula 4-4  = [  x 2 P (x )] - µ 2 Mean, Variance and Standard Deviation of a Probability Distribution

14. 14 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Formula 4-1 µ =  [x P (x)] Formula 4-2  2 =  [ (x - µ) 2 P (x )] Formula 4-3  2 = [  x 2 P (x )] - µ 2 (shortcut) Formula 4-4  = [  x 2 P (x )] - µ 2 Mean, Variance and Standard Deviation of a Probability Distribution

15. 15 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Roundoff 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. 16 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Definition Expected Value The average value of outcomes E =  [ x P( x )]

17. 17 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman E =  [x P(x)] Event Win Lose

18. 18 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman E =  [x P(x)] Event Win Lose x $499 - $1

19. 19 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman E =  [x P(x)] Event Win Lose x $499 - $1 P(x) 0.001 0.999

20. 20 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman E =  [x P(x)] Event Win Lose x $499 - $1 P(x) 0.001 0.999 x P(x) 0.499 - 0.999

21. 21 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman E =  [x P(x)] Event Win Lose x $499 - $1 P(x) 0.001 0.999 x P(x) 0.499 - 0.999 E = -$.50