Known Probability Distributions

1. Known Probability Distributions • Engineers frequently work with data that can be modeled as one of several known probability distributions. • Being able to model the data allows us to: • model real systems • design • predict results • Key discrete probability distributions include: • binomial / multinomial • negative binomial • hypergeometric • Poisson

2. Discrete Uniform Distribution • Simplest of all discrete distributions • All possible values of the random variable have the same probability, i.e., f(x; k) = 1/ k, x = x1 , x2 , x3 , … , xk • Expectations of the discrete uniform distribution

3. Binomial & Multinomial Distributions • Bernoulli Trials • Inspect tires coming off the production line. Classify each as defective or not defective. Define “success” as defective. If historical data shows that 95% of all tires are defect-free, then P(“success”) = 0.05. • Signals picked up at a communications site are either incoming speech signals or “noise.” Define “success” as the presence of speech. P(“success”) = P(“speech”) • Administer a test drug to a group of patients with a specific condition. P(“success”) = ___________ • Bernoulli Process • n repeated trials • the outcome may be classified as “success” or “failure” • the probability of success (p) is constant from trial to trial • repeated trials are independent.

4. Binomial Distribution • Example: Historical data indicates that 10% of all bits transmitted through a digital transmission channel are received in error. Let X = the number of bits in error in the next 4 bits transmitted. Assume that the transmission trials are independent. What is the probability that • Exactly 2 of the bits are in error? • At most 2 of the 4 bits are in error? • more than 2 of the 4 bits are in error? • The number of successes, X, in n Bernoulli trials is called a binomial random variable.

5. Binomial Distribution • The probability distribution is called the binomial distribution. • b(x; n, p) = , x = 0, 1, 2, …, n where p = _________________ q = _________________ • For our example, • b(x; n, p) = _________________

6. For Our Example … • What is the probability that exactly 2 of the bits are in error? • At most 2 of the 4 bits are in error?

7. Your turn … • What is the probability that more than 2 of the 4 bits are in error?

8. Expectations of the Binomial Distribution • The mean and variance of the binomial distribution are given by μ =np σ2 = npq • Suppose, in our example, we check the next 20 bits. What are the expected number of bits in error? What is the standard deviation? μ =___________ σ2 =__________ , σ =__________

9. Another example A worn machine tool produces 1% defective parts. If we assume that parts produced are independent, what is the mean number of defective parts that would be expected if we inspect 25 parts? What is the expected variance of the 25 parts?

10. Helpful Hints … • Sometimes it helps to draw a picture. Suppose we inspect the next 5 parts … P(at least 3)  P(2 ≤ X ≤ 4)  P(less than 4)  • Appendix Table A.1 (pp. 742-747) lists Binomial Probability Sums, ∑rx=0b(x; n, p)

11. Your turn … • Use Table A.1 to determine 1. b(x; 15, 0.4) , P(X≤ 8) = ______________ 2. b(x; 15, 0.4) , P(X< 8) = ______________ 3. b(x; 12, 0.2) , P(2 ≤ X≤ 5) = ___________ 4. b(x; 4, 0.1) , P(X> 2) = ______________

12. Multinomial Experiments • What if there are more than 2 possible outcomes? (e.g., acceptable, scrap, rework) • That is, suppose we have: • n independent trials • k outcomes that are • mutually exclusive (e.g., ♠, ♣, ♥, ♦) • exhaustive (i.e., ∑all k pi = 1) • Then f(x1, x2, …, xk; p1, p2, …, pk, n) =

13. Example • Look at problem 5.22, pg. 152 f( __, __, __; ___, ___, ___, __) =_________________ = __________________________________

14. Hypergeometric Distribution • Example*: Automobiles arrive in a dealership in lots of 10. Five out of each 10 are inspected. For one lot, it is know that 2 out of 10 do not meet prescribed safety standards. What is probability that at least 1 out of the 5 tested from that lot will be found not meeting safety standards? *from Complete Business Statistics, 4th ed (McGraw-Hill)

15. This example follows a hypergeometric distribution: • A random sample of size n is selected without replacement from N items. • k of the N items may be classified as “successes” and N-k are “failures.” • The probability associated with getting x successes in the sample (given k successes in the lot.) Where, k = number of “successes” = 2 n = number in sample = 5 N = the lot size = 10 x = number found = 1 or 2

16. Hypergeometric Distribution • In our example, = _____________________________

17. Expectations of the Hypergeometric Distribution • The mean and variance of the hypergeometric distribution are given by • What are the expected number of cars that fail inspection in our example? What is the standard deviation? μ =___________ σ2 =__________ , σ =__________

18. Your turn … A worn machine tool produced defective parts for a period of time before the problem was discovered. Normal sampling of each lot of 20 parts involves testing 6 parts and rejecting the lot if 2 or more are defective. If a lot from the worn tool contains 3 defective parts: • What is the expected number of defective parts in a sample of six from the lot? • What is the expected variance? • What is the probability that the lot will be rejected?

19. Binomial Approximation • Note, if N >> n, then we can approximate this with the binomial distribution. For example: Automobiles arrive in a dealership in lots of 100. 5 out of each 100 are inspected. 2 /10 (p=0.2) are indeed below safety standards. What is probability that at least 1 out of 5 will be found not meeting safety standards? • Recall: P(X≥ 1) = 1 – P(X< 1) = 1 – P(X = 0) (Compare to example 5.15, pg. 155)

20. Negative Binomial Distribution • Example: Historical data indicates that 30% of all bits transmitted through a digital transmission channel are received in error. An engineer is running an experiment to try to classify these errors, and will start by gathering data on the first 10 errors encountered. What is the probability that the 10th error will occur on the 25th trial?

21. This example follows a negative binomial distribution: • Repeated independent trials. • Probability of success = p and probability of failure = q = 1-p. • Random variable, X, is the number of the trial on which the kth success occurs. • The probability associated with the kth success occurring on trial x is given by, Where, k = “success number” = 10 x = trial number on which k occurs = 25 p = probability of success (error) = 0.3 q = 1 – p= 0.7

22. Negative Binomial Distribution • In our example, = _____________________________

23. Geometric Distribution • Example: In our example, what is the probability that the 1st bit received in error will occur on the 5th trial? • This is an example of the geometric distribution, which is a special case of the negative binomial in which k = 1. • The probability associated with the 1st success occurring on trial x is given by = __________________________________

24. Your turn … A worn machine tool produces 1% defective parts. If we assume that parts produced are independent: • What is the probability that the 2nd defective part will be the 6th one produced? • What is the probability that the 1st defective part will be seen before 3 are produced? • How many parts can we expect to produce before we see the 1st defective part? (Hint: see Theorem 5.4, pg. 161)

25. Poisson Process • The number of occurrences in a given interval or region with the following properties: • “memoryless” • P(occurrence) during a very short interval or small region is proportional to the size of the interval and doesn’t depend on number occurring outside the region or interval. • P(X>1) in a very short interval is negligible

26. Poisson Process • Examples: • Number of bits transmitted per minute. • Number of calls to customer service in an hour. • Number of bacteria in a given sample. • Number of hurricanes per year in a given region.

27. Poisson Process • Example An average of 2.7 service calls per minute are received at a particular maintenance center. The calls correspond to a Poisson process. To determine personnel and equipment needs to maintain a desired level of service, the plant manager needs to be able to determine the probabilities associated with numbers of service calls. What is the probability that fewer than 2 calls will be received in any given minute?

28. Poisson Distribution • The probability associated with the number of occurrences in a given period of time is given by, Where, λ= average number of outcomes per unit time or region = 2.7 t = time interval or region = 1 minute

29. Our Example • The probability that fewer than 2 calls will be received in any given minute is … P(X < 2) =P(X = 0) + P(X = 1) = __________________________ • The mean and variance are both λt, so μ = _____________________ • Note: Table A.2, pp. 748-750, gives Σtp(x;μ)

30. Poisson Distribution • If more than 6 calls are received in a 3-minute period, an extra service technician will be needed to maintain the desired level of service. What is the probability of that happening? μ = λt = _____________________ P(X > 6) = 1 – P(X <6) = _____________________

31. Poisson Distribution

32. Poisson Distribution The effect of λ on the Poisson distribution

33. Continuous Probability Distributions • Many continuous probability distributions, including: • Uniform • Normal • Gamma • Exponential • Chi-Squared • Lognormal • Weibull

34. Uniform Distribution • Simplest – characterized by the interval endpoints, A and B. A ≤ x ≤ B = 0 elsewhere • Mean and variance: and

35. Example A circuit board failure causes a shutdown of a computing system until a new board is delivered. The delivery time X is uniformly distributed between 1 and 5 days. What is the probability that it will take 2 or more days for the circuit board to be delivered?

36. Normal Distribution • The “bell-shaped curve” • Also called the Gaussian distribution • The most widely used distribution in statistical analysis • forms the basis for most of the parametric tests we’ll perform later in this course. • describes or approximates most phenomena in nature, industry, or research • Random variables (X) following this distribution are called normal random variables. • the parameters of the normal distribution are μand σ(sometimes μand σ2.)

37. (μ = 5, σ = 1.5) Normal Distribution • The density function of the normal random variable X, with mean μ and variance σ2, is all x.

38. Standard Normal RV … • Note: the probability of X taking on any value between x1 and x2 is given by: • To ease calculations, we define a normal random variable where Z is normally distributed with μ = 0 and σ2= 1

39. Standard Normal Distribution • Table A.3: “Areas Under the Normal Curve”

40. Examples • P(Z ≤ 1) = • P(Z ≥ -1) = • P(-0.45 ≤ Z ≤ 0.36) =

41. Your turn … • Use Table A.3 to determine (draw the picture!) 1. P(Z≤ 0.8) = 2. P(Z≥ 1.96) = 3. P(-0.25 ≤ Z≤ 0.15) = 4. P(Z ≤ -2.0 orZ≥ 2.0) =

42. The Normal Distribution “In Reverse” • Example: Given a normal distribution with μ = 40 and σ = 6, find the value of X for which 45% of the area under the normal curve is to the left of X. • If P(Z < k) = 0.45, k = ___________ • Z = _______ X = _________

43. Normal Approximation to the Binomial • If n is large and p is not close to 0 or 1, or if n is smaller but p is close to 0.5, then the binomial distribution can be approximated by the normal distribution using the transformation: • NOTE: add or subtract 0.5 from X to be sure the value of interest is included (draw a picture to know which) • Look at example 6.15, pg. 191

44. Look at example 6.15, pg. 191 p = 0.4 n = 100 μ= ____________ σ= ______________ if x = 30, then z = _____________________ and, P(X < 30) = P (Z < _________) = _________

45. Your Turn DRAW THE PICTURE!! • Refer to the previous example, • What is the probability that more than 50 survive? • What is the probability that exactly 45 survive?

46. Gamma & Exponential Distributions • Recall the Poisson Process • Number of occurrences in a given interval or region • “Memoryless” process • Sometimes we’re interested in the time or area until a certain number of events occur. • For example An average of 2.7 service calls per minute are received at a particular maintenance center. The calls correspond to a Poisson process. • What is the probability that up to a minute will elapse before 2 calls arrive? • How long before the next call?

47. Gamma Distribution • The density function of the random variable X with gamma distribution having parameters α (number of occurrences) and β (time or region). x > 0. μ = αβ σ2= αβ2

48. Exponential Distribution • Special case of the gamma distribution with α = 1. x > 0. • Describes the time until or time between Poisson events. μ = β σ2= β2

49. Example An average of 2.7 service calls per minute are received at a particular maintenance center. The calls correspond to a Poisson process. What is the probability that up to a minute will elapse before 2 calls arrive? β = ________ α = ________ P(X ≤ 1) = _________________________________

50. Example (cont.) What is the expected time before the next call arrives? β = ________ α = ________ μ = _________________________________