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Overview

Overview. This chapter will deal with the construction of probability distributions Probability Distributions will describe what will probably happen instead of what actually did happen. Random Variables. Definitions. Random Variable

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  1. Overview This chapter will deal with the construction of probability distributions Probability Distributions will describe what will probably happen instead of what actually did happen.

  2. Random Variables

  3. 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

  4. 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.

  5. Requirements for Probability Distribution  P(x) = 1 where x assumes all possible values 0 P(x) 1 for every value of x

  6. Mean, Variance and Standard Deviation of a Probability Distribution Formula 6-1 µ = [x•P(x)] Formula 6-2 2= [(x - µ)2 • P(x)]

  7. .93 .04 .02 .01 $50000 $100000 $200000 0 The probability that a car insurance company pays $200,000 is .01, $100,000 is .02 and $50,000 is .04. • Draw the probability density function. • Determine the minimum value the insurance • company should charge to break even. µA= ($200000)(.01)+($100000)(.02)+($50000)(.04)+(0)(.93) = $6000

  8. .70 .12 0 .09 .07 .02 10 25 60 90 Page 243 #5 • Draw the probability density function. • Determine the minimum value the insurance • company should charge to break even. µA= ($10000)(.12)+($25000)(.09)+($60000)(.07)+ (.022) ($90000)+ (0)(.93) = $9450

  9. Definition Expected Value The average value of outcomes E =  [x • P(x)]

  10. What is the expected value for a $1.00 ticket that pays off a $500.00 prize if 1000 tickets are sold? E =  [x • P(x)] Event Win Lose

  11. E =  [x • P(x)] Event Win Lose x $499 - $1

  12. E =  [x • P(x)] Event Win Lose x $499 - $1 P(x) 0.001 0.999

  13. E =  [x • P(x)] Event Win Lose x $499 - $1 P(x) 0.001 0.999 x • P(x) 0.499 - 0.999

  14. 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

  15. Definitions • Binomial Probability Distribution 1. The experiment must have a fixed number of trials. • The trials must be independent. (The outcome of any individual trial doesn’t affect the probabilities in the other trials.) 3. Each trial must have all outcomes classified into two categories. 4. The probabilities must remain constant for each trial.

  16. Notation for Binomial Probability Distributions n = fixed number of trials r = specific number of successes in n trials p = probability of success in one of n trials q = probability of failure in one of n trials (q = 1 - p) P(r) = probability of getting exactly r success among n trials Be sure that r and p both refer to the same category being called a success.

  17. Binomial Probability Formula

  18. Binomial Probability Formula n! • P(x) = • pr• qn-r (n - r )! r!

  19. Method 1 Binomial Probability Formula n! • P(r) = • pr• qn-r (n - r )! r! • P(r) = nCr• pr•qn-r for calculators with nCr key.

  20. Example: Find the probability of getting exactly 3 correct responses among 5 different requests from AT&T directory assistance. Assume in general, AT&T is correct 90% of the time. This is a binomial experiment where: n = 5 r = 3 p = 0.90 q = 0.10

  21. Example: Find the probability of getting exactly 3 correct responses among 5 different requests from AT&T directory assistance. Assume in general, AT&T is correct 90% of the time. This is a binomial experiment where: n = 5 r = 3 p = 0.90 q = 0.10 Using the binomial probability formula to solve: P(3) = 5C3• 0.93• 0.12 =.0729

  22. Example: Using n = 5 and p = 0.90, find the following: a) The probability of exactly 3 successesb) The probability of at least 3 successes a) P(3) = 0.073 b) P(at least 3) = P(3 or 4 or 5) = P(3) or P(4) or P(5) = 0.073 + 0.328 + 0.590 = 0.991

  23. For Binomial Distributions: • Formula 4-6 µ = n • p • Formula 4-7 2 = n • p • q or = n • p • q

  24. Example: Find the mean and standard deviation for the number of girls in groups of 14 births. • n = 14 • p = 0.5 • q = 0.5 • Using the binomial distribution formulas:

  25. Example: Find the mean and standard deviation for the number of girls in groups of 14 births. • n = 14 • p = 0.5 • q = 0.5 • Using the binomial distribution formulas: • µ = (14)(0.5) = 7 girls = 1.9 girls

  26. Reminder • Maximum usual values = µ + 2  • Minimum usual values = µ - 2 

  27. Example: Determine whether 68 girls among 100 babies could easily occur by chance. • For this binomial distribution, • µ = 50 girls • = 5 girls • µ + 2 = 50 + 2(5) = 60 • µ - 2 = 50 - 2(5) = 40 • The usual number girls among 100 births would be from 40 to 60. So 68 girls in 100 births is an unusual result.

  28. Poisson Random Variable • Poisson process is the random occurrence • of a series of events over time (or space) in • such a way that: • The number of successes can be any whole number. • The probability of success is equally spread out. • the mean number of successes r the desired number of successes

  29. Example: On an average day, 6 people visit the barbershop • What is the probability that exactly 4 will visit the barbershop? • What is the probability that 2 or less will visit the barbershop? • What is the probability that more than 3 will visit the barbershop?

  30. Example: A 100-foot of pipe has 20 leaks. A 25-foot section is chosen at random • What is the probability it has exactly 3 leaks? • What is the probability it has at most 2 leaks? • What is the probability it has more than 3 leaks?  = 5

  31. Overview • Continuous random variable • Normal distribution

  32. Curve is bell shaped and symmetric µ Score Overview • Continuous random variable • Normal distribution

  33. Curve is bell shaped and symmetric µ Score Overview • Continuous random variable • Normal distribution x - µ2 ( ) 1  2 y = e 2 p

  34. Definitions • Uniform Distribution a probability distribution in which the continuous random variable values are spread evenly over the range of      possibilities; the graph results in a      rectangular shape.

  35. Definitions • Density Curve (or probability density function)   the graph of a continuous probability   distribution

  36. Definitions • Density Curve (or probability density function)   the graph of a continuous probability   distribution 1. The total area under the curve must equal 1. 2. Every point on the curve must have a vertical height that is 0 or greater.

  37. Because the total area under the density curve is equal to 1, there is a correspondence between area and probability.

  38. Times in First or Last Half Hours

  39. 2 10 A continuous random variable,T, has a uniform (rectangular) probability distribution over the interval [2,10]. • Graph the probability distribution. • Find P(x>6) • Find P(5<x<8) • A = b*h, 1=b*8, h = .125 • P(x>6) A = b*h = 4*.125 = .5 • P(5<x<8) A = b*h = 3*.125 = .375

  40. 0 6 A continuous random variable,T, has a uniform (triangular) probability distribution over the interval [0,6]. (0, 1/3) (1, 5/18) (2, 2/9) • Graph the probability distribution. • Find P(x > 4) • Find P(1 < x < 4) (3, 1/6) (4, 1/9) (5, 1/18) • A = ½ b*h, 1= ½* 6*h, h = 1/3 • P(x > 4) A = ½ b*h = ½ *2*(1/9) = 1/9 • P(1 < x < 4) A = ½ b1*h1 - ½ b2*h2 = • ½* 5*(5/18) – ½ *2*(1/9) = 25/36 – 1/9 = 21/36 = • 7/12

  41. Heights of Adult Men and Women Women: µ = 63.6  = 2.5 Men: µ = 69.0  = 2.8 63.6 69.0 Figure 5-4 Height (inches)

  42. Definition Standard Normal Deviation a normal probability distribution that has a mean of 0 and a standard deviation of 1

  43. Table A-2 • Formulas and Tables card • Appendix A2

  44. Table A-2 Standard Normal Distribution µ = 0  = 1 0 x z

  45. -3 -2 -1 0 1 2 3 Definition Standard Normal Deviation a normal probability distribution that has a mean of 0 and a standard deviation of 1 Area found in Table A-2 Area = 0.3413 0.4429 z = 1.58 0 Score (z ) Figure 5-6 Figure 5-5

  46. Table A-2 Standard Normal (z) Distribution z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 .0000 .0398 .0793 .1179 .1554 .1915 .2257 .2580 .2881 .3159 .3413 .3643 .3849 .4032 .4192 .4332 .4452 .4554 .4641 .4713 .4772 .4821 .4861 .4893 .4918 .4938 .4953 .4965 .4974 .4981 .4987 .0040 .0438 .0832 .1217 .1591 .1950 .2291 .2611 .2910 .3186 .3438 .3665 .3869 .4049 .4207 .4345 .4463 .4564 .4649 .4719 .4778 .4826 .4864 .4896 .4920 .4940 .4955 .4966 .4975 .4982 .4987 .0080 .0478 .0871 .1255 .1628 .1985 .2324 .2642 .2939 .3212 .3461 .3686 .3888 .4066 .4222 .4357 .4474 .4573 .4656 .4726 .4783 .4830 .4868 .4898 .4922 .4941 .4956 .4967 .4976 .4982 .4987 .0120 .0517 .0910 .1293 .1664 .2019 .2357 .2673 .2967 .3238 .3485 .3708 .3907 .4082 .4236 .4370 .4484 .4582 .4664 .4732 .4788 .4834 .4871 .4901 .4925 .4943 .4957 .4968 .4977 .4983 .4988 .0160 .0557 .0948 .1331 .1700 .2054 .2389 .2704 .2995 .3264 .3508 .3729 .3925 .4099 .4251 .4382 .4495 .4591 .4671 .4738 .4793 .4838 .4875 .4904 .4927 .4945 .4959 .4969 .4977 .4984 .4988 .0199 .0596 .0987 .1368 .1736 .2088 .2422 .2734 .3023 .3289 .3531 .3749 .3944 .4115 .4265 .4394 .4505 .4599 .4678 .4744 .4798 .4842 .4878 .4906 .4929 .4946 .4960 .4970 .4978 .4984 .4989 .0239 .0636 .1026 .1406 .1772 .2123 .2454 .2764 .3051 .3315 .3554 .3770 .3962 .4131 .4279 .4406 .4515 .4608 .4686 .4750 .4803 .4846 .4881 .4909 .4931 .4948 .4961 .4971 .4979 .4985 .4989 .0279 .0675 .1064 .1443 .1808 .2157 .2486 .2794 .3078 .3340 .3577 .3790 .3980 .4147 .4292 .4418 .4525 .4616 .4693 .4756 .4808 .4850 .4884 .4911 .4932 .4949 .4962 .4972 .4979 .4985 .4989 .0319 .0714 .1103 .1480 .1844 .2190 .2517 .2823 .3106 .3365 .3599 .3810 .3997 .4162 .4306 .4429 .4535 .4625 .4699 .4761 .4812 .4854 .4887 .4913 .4934 .4951 .4963 .4973 .4980 .4986 .4990 .0359 .0753 .1141 .1517 .1879 .2224 .2549 .2852 .3133 .3389 .3621 .3830 .4015 .4177 .4319 .4441 .4545 .4633 .4706 .4767 .4817 .4857 .4890 .4916 .4936 .4952 .4964 .4974 .4981 .4986 .4990 * *

  47. To find:zScore the distance along horizontal scale of the standard normal distribution; refer to the leftmost column and top rowofTable A-2 Area the region under the curve; refer to the values in the body of Table A-2

  48. Example: If thermometers have an average (mean) reading of 0 degrees and a standard deviation of 1 degree for freezing water and if one thermometer is randomly selected, find the probability that it reads freezing water between 0 degrees and 1.58 degrees.

  49. Example: If thermometers have an average (mean) reading of 0 degrees and a standard deviation of 1 degree for freezing water and if one thermometer is randomly selected, find the probability that it reads freezing water between 0 degrees and 1.58 degrees. P ( 0 < x < 1.58 ) = 0 1.58

  50. Table A-2 Standard Normal (z) Distribution z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 .0000 .0398 .0793 .1179 .1554 .1915 .2257 .2580 .2881 .3159 .3413 .3643 .3849 .4032 .4192 .4332 .4452 .4554 .4641 .4713 .4772 .4821 .4861 .4893 .4918 .4938 .4953 .4965 .4974 .4981 .4987 .0040 .0438 .0832 .1217 .1591 .1950 .2291 .2611 .2910 .3186 .3438 .3665 .3869 .4049 .4207 .4345 .4463 .4564 .4649 .4719 .4778 .4826 .4864 .4896 .4920 .4940 .4955 .4966 .4975 .4982 .4987 .0080 .0478 .0871 .1255 .1628 .1985 .2324 .2642 .2939 .3212 .3461 .3686 .3888 .4066 .4222 .4357 .4474 .4573 .4656 .4726 .4783 .4830 .4868 .4898 .4922 .4941 .4956 .4967 .4976 .4982 .4987 .0120 .0517 .0910 .1293 .1664 .2019 .2357 .2673 .2967 .3238 .3485 .3708 .3907 .4082 .4236 .4370 .4484 .4582 .4664 .4732 .4788 .4834 .4871 .4901 .4925 .4943 .4957 .4968 .4977 .4983 .4988 .0160 .0557 .0948 .1331 .1700 .2054 .2389 .2704 .2995 .3264 .3508 .3729 .3925 .4099 .4251 .4382 .4495 .4591 .4671 .4738 .4793 .4838 .4875 .4904 .4927 .4945 .4959 .4969 .4977 .4984 .4988 .0199 .0596 .0987 .1368 .1736 .2088 .2422 .2734 .3023 .3289 .3531 .3749 .3944 .4115 .4265 .4394 .4505 .4599 .4678 .4744 .4798 .4842 .4878 .4906 .4929 .4946 .4960 .4970 .4978 .4984 .4989 .0239 .0636 .1026 .1406 .1772 .2123 .2454 .2764 .3051 .3315 .3554 .3770 .3962 .4131 .4279 .4406 .4515 .4608 .4686 .4750 .4803 .4846 .4881 .4909 .4931 .4948 .4961 .4971 .4979 .4985 .4989 .0279 .0675 .1064 .1443 .1808 .2157 .2486 .2794 .3078 .3340 .3577 .3790 .3980 .4147 .4292 .4418 .4525 .4616 .4693 .4756 .4808 .4850 .4884 .4911 .4932 .4949 .4962 .4972 .4979 .4985 .4989 .0319 .0714 .1103 .1480 .1844 .2190 .2517 .2823 .3106 .3365 .3599 .3810 .3997 .4162 .4306 .4429 .4535 .4625 .4699 .4761 .4812 .4854 .4887 .4913 .4934 .4951 .4963 .4973 .4980 .4986 .4990 .0359 .0753 .1141 .1517 .1879 .2224 .2549 .2852 .3133 .3389 .3621 .3830 .4015 .4177 .4319 .4441 .4545 .4633 .4706 .4767 .4817 .4857 .4890 .4916 .4936 .4952 .4964 .4974 .4981 .4986 .4990 * *

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