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
• s2 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 ___________
• s2 __________ , s __________

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, ?rx0b(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 lt 8) ______________
• 3. b(x 12, 0.2) , P(2 X 5) ___________
• 4. b(x 4, 0.1) , P(X gt 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( __, __, __ ___, ___, ___, __)
  _________________
• __________________________________

x1 _______ p1 _______
x2 _______ p2 _______ n _____
x3 _______ p3 _______
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 ___________
• s2 __________ , s __________

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 gtgt 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
  (p0.2) 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 lt 1) 1 P(X 0)

Hypergeometric distribution Binomial distribution

(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 doesnt depend on number occurring
  outside the region or interval.
• P(Xgt1) 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,
• ? 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 lt 2) P(X 0) P(X 1)
• __________________________
• The mean and variance are both ?t, so
• µ _____________________
• Note Table A.2, pp. 748-750, gives St p(xµ)

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 ?t _____________________
• P(X gt 6) 1 P(X lt 6)
• _____________________

31
Poisson Distribution
32
Poisson Distribution

• The effect of ? 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
  well 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 µ
  and s (sometimes µ and s2.)

37
Normal Distribution

• The density function of the normal random
  variable X, with mean µ and variance s2, 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
  s2 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 or Z 2.0)

42
The Normal Distribution In Reverse

• Example
• Given a normal distribution with µ 40 and s
  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 lt 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 ____________ s ______________
• if x 30, then z _____________________
• and, P(X lt 30) P (Z lt _________) _________

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 were 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 a
  (number of occurrences) and ß (time or region).
• x gt 0.
• µ aß
• s2 aß2

48
Exponential Distribution

• Special case of the gamma distribution with a
  1.
• x gt 0.
• Describes the time until or time between Poisson
  events.
• µ ß
• s2 ß2

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?
• ß ________ a ________
• P(X 1) _________________________________

50
Example (cont.)

• What is the expected time before the next call
  arrives?
• ß ________ a ________
• µ _________________________________

51
Your turn

• Look at problem 6.40, page 205.

52
Chi-Squared Distribution

• Special case of the gamma distribution with a
  ?/2 and ß 2.
• x gt 0.
• where ? is a positive integer.
• single parameter,? is called the degrees of
  freedom.
• µ ?
• s2 2?

EGR 252 Ch. 6
52
53
Lognormal Distribution

• When the random variable Y ln(X) is normally
  distributed with mean µ and standard deviation s,
  then X has a lognormal distribution with the
  density function,

EGR 252 Ch. 6
53
54
Example

• Look at problem 6.72, pg. 207
• Since ln(X) has normal distribution with µ 5
  and s 2, the probability that X gt 50,000 is,
• P(X gt 50,000) __________________________

EGR 252 Ch. 6
54
55
Wiebull Distribution

• Used for many of the same applications as the
  gamma and exponential distributions, but
• does not require memoryless property of the
  exponential

EGR 252 Ch. 6
55
56
Example

• Designers of wind turbines for power generation
  are interested in accurately describing
  variations in wind speed, which in a certain
  location can be described using the Weibull
  distribution with a 0.02 and ß 2. A
  designer is interested in determining the
  probability that the wind speed in that location
  is between 3 and 7 mph.
• P(3 lt X lt 7) ___________________________

EGR 252 Ch. 6
56
57
Populations and Samples

• Population a group of individual persons,
  objects, or items from which samples are taken
  for statistical measurement
• Sample a finite part of a statistical
  population whose properties are studied to gain
  information about the whole

(Merriam-Webster Online Dictionary,
http//www.m-w.com/, October 5, 2004)
58
Examples

• Population
• Students pursuing undergraduate engineering
  degrees
• Cars capable of speeds in excess of 160 mph.
• Potato chips produced at the Frito-Lay plant in
  Kathleen
• Freshwater lakes and rivers

• Samples

59
Basic Statistics (review)

• 1. Sample Mean
• Example
• At the end of a team project, team members were
  asked to give themselves and each other a grade
  on their contribution to the group. The results
  for two team members were as follows
• ___________________
• ___________________

Q S
92 85
95 88
85 75
78 92
60
Basic Statistics (review)

• 1. Sample Variance
• For our example
• SQ2 ___________________
• SS2 ___________________

Q S
92 85
95 88
85 75
78 92
61
Your Turn

• Work in groups of 4 or 5. Find the mean,
  variance, and standard deviation for your group
  of the (approximate) number of hours spent
  working on homework each week.

62
Sampling Distributions

• If we conduct the same experiment several times
  with the same sample size, the probability
  distribution of the resulting statistic is called
  a sampling distribution
• Sampling distribution of the mean if n
  observations are taken from a normal population
  with mean µ and variance s2, then

63
Central Limit Theorem

• Given
• X the mean of a random sample of size n taken
  from a population with mean µ and finite variance
  s2,
• Then,
• the limiting form of the distribution of
• is _________________________

64
Central Limit Theorem

• If the population is known to be normal, the
  sampling distribution of X will follow a normal
  distribution.
• Even when the distribution of the population is
  not normal, the sampling distribution of X is
  normal when n is large.
• NOTE when n is not large, we cannot assume the
  distribution of X is normal.

65
Example

• The time to respond to a request for information
  from a customer help line is uniformly
  distributed between 0 and 2 minutes. In one month
  48 requests are randomly sampled and the response
  time is recorded.
• What is the probability that the average
  response time is between 0.9 and 1.1 minutes?
• µ ______________ s2 ________________
• µX __________ sX2 ________________
• Z1 _____________ Z2 _______________
• P(0.9 lt X lt 1.1) _____________________________

66
Sampling Distribution of the Difference Between
two Averages

• Given
• Two samples of size n1 and n2 are taken from two
  populations with means µ1 and µ2 and variances
  s12 and s22
• Then,

67
Sampling Distribution of S2

• Given
• S2 is the variance of of a random sample of size
  n taken from a population with mean µ and finite
  variance s2,
• Then,
• has a ?2 distribution with ? n - 1

68
?2 Distribution

• ?a2 represents the ?2 value above which we find
  an area of a, that is, for which P(?2 gt ?a2 ) a.

69
Example

• Look at example 8.10, pg. 256
• µ 3 s 1 n 5
• s2 ________________
• ?2 __________________
• If the ?2 value fits within an interval that
  covers 95 of the ?2 values with 4 degrees of
  freedom, then the estimate for s is reasonable.
• (See Table A.5, pp. 755-756)

70
Your turn

• If a sample of size 7 is taken from a normal
  population (i.e., n 7), what value of ?2
  corresponds to P(?2 lt ?a2) 0.95? (Hint first
  determine a.)

71
t- Distribution

• Recall, by CLT
• is n(z 0,1)
• Assumption _____________________
• (Generally, if an engineer is concerned with a
  familiar process or system, this is reasonable,
  but )

72
What if we dont know s?

• New statistic
• Where,
• and
• follows a t-distribution with ? n 1 degrees
  of freedom.

73
Characteristics of the t-Distribution

• Look at fig. 8.13, pg. 259
• Note
• Shape _________________________
• Effect of ? __________________________
• See table A.4, pp. 753-754

74
Using the t-Distribution

• Testing assumptions about the value of µ
• Example problem 8.52, pg. 265
• What value of t corresponds to P(t lt ta) 0.95?

75
Comparing Variances of 2 Samples

• Given two samples of size n1 and n2, with sample
  means X1 and X2, and variances, s12 and s22
• Are the differences we see in the means due to
  the means or due to the variances (that is, are
  the differences due to real differences between
  the samples or variability within each samples)?
• See figure 8.16, pg. 262

76
F-Distribution

• Given
• S12 and S22, the variances of independent random
  samples of size n1 and n2 taken from normal
  populations with variances s12 and s22,
  respectively,
• Then,
• has an F-distribution with ?1 n1 - 1 and ?2
  n2 1 degrees of freedom.
• (See table A.6, pp. 757-760)

77
Example

• Problem 8.55, pg. 266
• S12 ___________________
• S22 ___________________
• F _____________ f0.05 (4, 5) _________
• NOTE