1 / 51

Agenda

Agenda. Week 12. Review Homework Chapter 6: 1, 2, 3, 4, 13 Chapter 7 - 2, 5, 11 Probability Control charts for attributes. Week 13 Assignment Read Chapter 10: “Reliability” Homework Chapter 8: 5, 9,10, 20, 26, 33, 34 Chapter 9: 9, 23. Probability . Chapter Eight.

zofia
Télécharger la présentation

Agenda

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Agenda Week 12 • Review Homework • Chapter 6: 1, 2, 3, 4, 13 • Chapter 7 - 2, 5, 11 • Probability • Control charts for attributes • Week 13 Assignment • Read Chapter 10: “Reliability” • Homework • Chapter 8: 5, 9,10, 20, 26, 33, 34 • Chapter 9: 9, 23

  2. Probability Chapter Eight Probability

  3. Probability theorems • Probability is expressed as a number between 0 and 1 • Sum of the probabilities of the events of a situation equals 1 • If P(A) is the probability that an event will occur, then the probability the event will not occur is • 1.0 - P(A) Probability

  4. Probability theorems • For mutually exclusive events, the probability that either event A or event B will occur is the the sum of their respective probabilities. • When events A and B are not mutually exclusive events, the probability that either event A or event B will occur is • P(A or B or both) = P(A) + P(B) - P(both) Probability

  5. Probability theorems • If A and B are dependent events, the probability that both A and B will occur is • P(A and B) = P(A) x P(B|A) • If A and B are independent events, then the probability that both A and B will occur is • P(A and B) = P(A) x P(B) Probability

  6. Permutations and combinations • A permutation is the number of arrangements that n objects can have when r of them are used. • When the order in which the items are used is not important, the number of possibilities can be calculated by using the formula for a combination. Probability

  7. Discrete probability distributions • Hypergeometric - random samples from small lot sizes. • Population must be finite • samples must be taken randomly without replacement • Binomial - categorizes “success” and “failure” trials • Poisson - quantifies the count of discrete events. Probability

  8. Continuous probability distributions • Normal • Uniform • Exponential • Chi Square • F • student t Probability

  9. Fundamental concepts • Probability = occurrences/trials • 0 < P < 1 • The sum of the simple probabilities for all possible outcomes must equal 1 • Complementary rule - P(A) + P(A’) = 1 Probability

  10. Addition rule • P(A + B) = P(A) + P(B) - P(A and B) • If mutually exclusive; just P(A) + P(B) P(AandB) P(A) P(B) Probability

  11. Addition rule example • P(A + B) = P(A) + P(B) - P(A and B) • Roll one die • Probability of even and divisible by 1.5? • Sample space {1,2,3,4,5,6} • Event A - Even {2,4,6} • Event B - Divisible by 1.5 {3,6} • Event A and B ? • Solution? Probability

  12. Conditional probability rule • P(A|B) = P(A and B) / P(B) • A die is thrown and the result is known to be an even number. What is the probability that this number is divisible by 1.5? • P(/1.5|Even)=P(/1.5 and even)/P(even) • 1/6 / 3/6 = 1/3 Probability

  13. Compound or joint probability • The probability of the simultaneous occurrence of two or more events is called the compound probability or, synonymously, the joint probability. • Mutually exclusive events cannot be independent unless one of them is zero. Probability

  14. Multiplication for independent events • P(A and B) = P(A) x P(B) • P(ace and heart) = P(ace) x P(heart) • 1/13 x 1/4 = 1/52 Probability

  15. Computing conditional probabilities • P(A|B) = P(A and B)/P(B) • P(Own and Less than 2 years)? Probability

  16. Computing conditional probabilities • P(A|B) = P(A and B)/P(B) P(AandB) P(A) P(B) Probability

  17. Probability

  18. Conditional probability Satisfied Not Satisfied Totals New 300 100 400 Used 450 150 600 Total 750 250 1000 S=satisfied N= bought new car P(N|S) = ? Probability

  19. 60 business students from a large university are surveyed with the following results: 19 read Business Week 18 read WSJ 50 read Fortune 13 read BW and WSJ 11 read WSJ and Fortune 13 read BW and Fortune 9 read all three How many read none? How many read only Fortune? How many read BW, the WSJ, but not Fortune? Hint: Try a Venn diagram. Just for fun Probability

  20. Probability Distributions Probability

  21. Learning objectives • Know the difference between discrete and continuous random variables. • Provide examples of discrete and continuous probability distributions. • Calculate expected values and variances. • Use the normal distribution table. Probability

  22. Random variables • A random variable is a numerical quantity whose value is determined by chance. • “A random variable assigns a number to every possible outcome or event in an experiment”. • For non-numerical outcomes such as a coin flip you must assign a random variable that associates each outcome with a unique real number. Probability

  23. Random variable types • Discrete random variable - assumes a limited set of values; non-continuous, generally countable • number of Mark McGwire homeruns in a season • number of auto parts passing assembly-line inspection • GRE exam scores Probability

  24. Random variable types • Continuous random variable - random variable with an infinite set of values. 0.000 Baseball player’s batting average 1.000 Can occur anywhere on a continuous number scale Probability

  25. Random variables and probability distributions • The relationship between a random variable’s values and their probabilities is summarized by its probability distribution. Probability

  26. Probability distribution • Whether continuous or discrete, the probability distribution provides a probability for each possible value of a random variable, and follows these rules: • The events are mutually exclusive • The individual probability values are between 0 and 1. • The total value of the probability values sum to 1 Probability

  27. Possible rate of return 10% 11% 12% 13% 14% 15% 16% 17% Probability .05 .15 .20 .35 .10 .10 .03 .02 Total = 1.0 Probability distribution for rates of return Probability

  28. Measures of central tendency expected value (weighted average) Measures of variability variance standard deviation Describing distributions Probability

  29. Expected value of a discrete random variable • For discrete random variables, the expected value is the sum of all the possible outcomes times the probability that they occur. E(X) = {xi * P(xi)} Probability

  30. Example: A fair die • Roll 1 die: x P(x) x*P(x) E(x)=? 1 1/6 1/6 2 1/6 2/6 3 1/6 3/6 4 1/6 4/6 5 1/6 5/6 6 1/6 6/6 Can you sketch the distribution? Probability

  31. Fair die illustrates a discrete “uniform distribution” • The random variable, x, has n possible outcomes and each outcome is equally likely. Thus, x is distributed uniform. Probability

  32. Probability distribution P(x) 1/6 x 1 2 3 4 5 6 Probability

  33. Example: An unfair die • Roll 1 die: x P(x) x*P(x) E(x)=? 1 1/12 1/12 2 2/12 4/12 3 2/12 6/12 4 2/12 8/12 5 2/12 10/12 6 3/12 18/12 Can you sketch the distribution? Probability

  34. Expected value of a bet • Suppose I offer you the following wager: You roll 1 die. If the result is even, I pay you $2.00. Otherwise you pay me $1.00. • E(your winnings)=.5 ($2.00) + .5 (-1.00) = 1.00 - .50 = $0.50 Probability

  35. Expected Value of a Bet • Suppose I offer you the following wager: You roll 1 die. If the result is 5 or 6 I pay you $3.00. Otherwise you pay me $2.00. • What is your expected value? Probability

  36. Variance of a discrete random variable The variance of a random variable is a measure of dispersion calculated by squaring the differences between the expected value and each random variable and multiplying by its associated probability. {(xi-E(x))2 * P(xi)} Probability

  37. Example: A fair die • Roll 1 die: [x- E(X)] 2 P(x) *P(x) 1 - 21/6 6.25 1/6 1.04 2 - 21/6 2.25 1/6 .375 3 - 21/6 .25 1/6 .04 4 - 21/6 .25 1/6 .04 5 - 21/6 2.25 1/6 .375 6 - 21/6 6.25 1/6 1.04 2.91 Probability

  38. Probability distributions for continuous random variables • A continuous mathematical function describes the probability distribution. • It’s called the probability density function and designated ƒ(x) • Some well know continuous probability density functions: • Normal Beta • Exponential Student t Probability

  39. Continuous probability density function - Uniform If a random variable, x, is distributed uniform over the interval [a,b], then its pdf is given by 1 b-a a b Probability

  40. Uniform What is the probability of x? 1 b-a x a b Probability

  41. Uniform Area under the rectangle = base*height = (b-a)* 1 = 1 b-a 1 b-a a b Probability

  42. Uniform P(c<x<b) = Area of brown rectangle 1 * (b-c) Ht x Width) = b-a 1 b-a c a b Probability

  43. Uniform P(2<x<5) = Brown rectangle 1 * (5-2) =(1/4) *3 =3/4 = 5-1 1 5-1 =1/4 2 1 5 Probability

  44. Uniform distribution If a random variable, x, is distributed uniform over the interval [a,b], then its pdf is given by And, the mean and variance are (a+b) ( b-a )2 E(x) = ------- Var(x)=--------- 2 12 Probability

  45. Uniform Mean? Variance? 3 8 Probability

  46. Calculate uniform mean, variance So, if a = 3 and b = 8 And, the mean and variance are (a+b) ( b-a )2 25 E(x) = ------ = 5.5 V(x)=--------- = ----- = 2.08 2 12 12 Probability

  47. Continuous pdf - Normal If x is a normally distributed variable, then is the pdf for x. The expected value is  and the variance is 2. Probability

  48. One standard deviation   68.3% Probability

  49. Two standard deviations 2 2 95.5% Probability

  50. Three standard deviations 3 3 99.73% Probability

More Related