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

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

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.

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.

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) _________________

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?

Your turn

- What is the probability that more than 2 of the 4

bits are in error?

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 __________

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?

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)

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) ______________

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)

Example

- Look at problem 5.22, pg. 152
- f( __, __, __ ___, ___, ___, __)

_________________ - __________________________________

x1 _______ p1 _______

x2 _______ p2 _______ n _____

x3 _______ p3 _______

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)

- 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

Hypergeometric Distribution

- In our example,
- _____________________________

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 __________

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?

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)

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?

- 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

Negative Binomial Distribution

- In our example,
- _____________________________

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 - __________________________________

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)

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

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.

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?

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

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)

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)
- _____________________

Poisson Distribution

Poisson Distribution

- The effect of ? on the Poisson distribution

Continuous Probability Distributions

- Many continuous probability distributions,

including - Uniform
- Normal
- Gamma
- Exponential
- Chi-Squared
- Lognormal
- Weibull

Uniform Distribution

- Simplest characterized by the interval

endpoints, A and B. - A x B
- 0 elsewhere
- Mean and variance
- and

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?

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.)

Normal Distribution

- The density function of the normal random

variable X, with mean µ and variance s2, is - all x.

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

Standard Normal Distribution

- Table A.3 Areas Under the Normal Curve

Examples

- P(Z 1)
- P(Z -1)
- P(-0.45 Z 0.36)

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)

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 _________

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

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 _________) _________

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?

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?

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

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

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) _________________________________

Example (cont.)

- What is the expected time before the next call

arrives? - ß ________ a ________
- µ _________________________________

Your turn

- Look at problem 6.40, page 205.

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

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

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

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

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

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)

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

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

Basic Statistics (review)

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

Q S

92 85

95 88

85 75

78 92

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.

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

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 _________________________

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.

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) _____________________________

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,

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

?2 Distribution

- ?a2 represents the ?2 value above which we find

an area of a, that is, for which P(?2 gt ?a2 ) a.

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)

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.)

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 )

What if we dont know s?

- New statistic
- Where,
- and
- follows a t-distribution with ? n 1 degrees

of freedom.

Characteristics of the t-Distribution

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

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?

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

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)

Example

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