![Page 1: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/1.jpg)
South Dakota
School of Mines & Technology
Introduction to Introduction to Probability & StatisticsProbability & Statistics
Industrial Engineering
![Page 2: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/2.jpg)
Introduction to Probability & Statistics
Concepts of ProbabilityConcepts of Probability
![Page 3: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/3.jpg)
Probability Concepts
S = Sample Space : the set of all possible unique outcomes of a repeatable experiment.
Ex: flip of a coin S = {H,T}
No. dots on top face of a dieS = {1, 2, 3, 4, 5, 6}
Body Temperature of a live humanS = [88,108]
![Page 4: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/4.jpg)
Probability Concepts
Event: a subset of outcomes from a sample space.
Simple Event: one outcome; e.g. get a 3 on one throw of a die
A = {3}
Composite Event: get 3 or more on throw of a die
A = {3, 4, 5, 6}
![Page 5: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/5.jpg)
Rules of Events
Union: event consisting of all outcomes present
in one or more of events making up
union.Ex:
A = {1, 2} B = {2, 4, 6}
A B = {1, 2, 4, 6}
![Page 6: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/6.jpg)
Rules of Events
Intersection: event consisting of all outcomes present in each contributing event.
Ex:A = {1, 2} B = {2, 4, 6}
A B = {2}
![Page 7: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/7.jpg)
Rules of Events
Complement: consists of the outcomes in the sample space which are not in stipulated event
Ex:A = {1, 2} S = {1, 2, 3, 4, 5,
6}
A = {3, 4, 5, 6}
![Page 8: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/8.jpg)
Rules of Events
Mutually Exclusive: two events are mutually exclusive if their intersection is null
Ex:A = {1, 2, 3} B = {4, 5, 6}
A B = { } =
![Page 9: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/9.jpg)
Probability Defined Equally Likely Events
If m out of the n equally likely outcomes in an experiment pertain to event A, then
p(A) = m/n
![Page 10: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/10.jpg)
Probability Defined Equally Likely Events
If m out of the n equally likely outcomes in an experiment pertain to event A, then
p(A) = m/n
Ex: Die example has 6 equally likely outcomes:p(2) = 1/6p(even) = 3/6
![Page 11: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/11.jpg)
Probability Defined
Suppose we have a workforce which is comprised of 6 technical people and 4 in administrative support.
![Page 12: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/12.jpg)
Probability Defined Suppose we have a workforce which is
comprised of 6 technical people and 4 in administrative support.
P(technical) = 6/10 P(admin) = 4/10
![Page 13: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/13.jpg)
Rules of Probability
Let A = an event defined on the event space S
1. 0 < P(A) < 12. P(S) = 13. P( ) = 04. P(A) + P( A ) = 1
![Page 14: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/14.jpg)
Addition Rule
P(A B) = P(A) + P(B) - P(A B)
A B
![Page 15: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/15.jpg)
Addition Rule
P(A B) = P(A) + P(B) - P(A B)
A B
![Page 16: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/16.jpg)
Example Suppose we have technical and
administrative support people some of whom are male and some of whom are female.
![Page 17: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/17.jpg)
Example (cont) If we select a worker at random, compute the following probabilities:
P(technical) = 18/30
![Page 18: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/18.jpg)
Example (cont) If we select a worker at random, compute the following probabilities:
P(female) = 14/30
![Page 19: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/19.jpg)
Example (cont) If we select a worker at random, compute the following probabilities:
P(technical or female) = 22/30
![Page 20: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/20.jpg)
Example (cont) If we select a worker at random, compute the following probabilities:
P(technical and female) = 10/30
![Page 21: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/21.jpg)
Alternatively we can find the probability of randomly selecting a technical person or a female by use of the addition rule.
= 18/30 + 14/30 - 10/30
= 22/30
Example (cont)
)()()()( FTPFPTPFTP -+=
![Page 22: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/22.jpg)
Operational Rules
Mutually Exclusive Events:
P(A B) = P(A) + P(B)
A B
![Page 23: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/23.jpg)
Conditional Probability
Now suppose we know that event A has occurred. What is the probability of B given A?
A A B
P(B|A) = P(A B)/P(A)
![Page 24: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/24.jpg)
Example
Returning to our workers, suppose we know we have a technical person.
![Page 25: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/25.jpg)
Example
Returning to our workers, suppose we know we have a technical person. Then, P(Female | Technical) = 10/18
![Page 26: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/26.jpg)
Example Alternatively,
P(F | T) = P(F T) / P(T) = (10/30) / (18/30) = 10/18
![Page 27: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/27.jpg)
Independent Events
Two events are independent if
P(A|B) = P(A)or
P(B|A) = P(B)
In words, the probability of A is in no way affected by the outcome of B or vice versa.
![Page 28: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/28.jpg)
Example
Suppose we flip a fair coin. The possible outcomes are
H T
The probability of getting a head is then
P(H) = 1/2
![Page 29: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/29.jpg)
Example
If the first coin is a head, what is the probability of getting a head on the second toss?
H,H H,TT,H T,T
P(H2|H1) = 1/2
![Page 30: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/30.jpg)
Example Suppose we flip a fair coin twice. The
possible outcomes are:
H,H H,TT,H T,T
P(2 heads) = P(H,H) = 1/4
![Page 31: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/31.jpg)
Example Alternatively
P(2 heads) = P(H1 H2)
= P(H1)P(H2|H1)
= P(H1)P(H2)
= 1/2 x 1/2
= 1/4
![Page 32: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/32.jpg)
Example Suppose we have a workforce consisting
of male technical people, female technical people, male administrative support, and female administrative support. Suppose the make up is as followsTech Admin
Male
Female
8
10
8
4
![Page 33: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/33.jpg)
Example
Let M = male, F = female, T = technical, and A = administrative. Compute the following:
P(M T) = ?
P(T|F) = ?
P(M|T) = ?
Tech Admin
Male
Female
8
10
8
4
![Page 34: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/34.jpg)
![Page 35: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/35.jpg)
South Dakota
School of Mines & Technology
Introduction to Introduction to Probability & StatisticsProbability & Statistics
Industrial Engineering
![Page 36: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/36.jpg)
Introduction to Probability & Statistics
CountingCounting
![Page 37: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/37.jpg)
Fundamental Rule
If an action can be performed in m ways and another action can be performed in n ways, then both actions can be performed in m•n ways.
![Page 38: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/38.jpg)
Fundamental Rule
Ex: A lottery game selects 3 numbers between 1 and 5 where numbers can not be selected more than once. If the game is truly random and order is not important, how many possible combinations of lottery numbers are there?
![Page 39: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/39.jpg)
Fundamental Rule Ex: A lottery game selects 3 numbers between 1 and 5
where numbers can not be selected more than once. If the game is truly random and order is not important, how many possible combinations of lottery numbers are there?
1
2
3
4
5
![Page 40: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/40.jpg)
Fundamental Rule Ex: A lottery game selects 3 numbers between 1 and 5
where numbers can not be selected more than once. If the game is truly random and order is not important, how many possible combinations of lottery numbers are there?
1
2
3
4
5
2345
![Page 41: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/41.jpg)
Fundamental Rule Ex: A lottery game selects 3 numbers between 1 and 5
where numbers can not be selected more than once. If the game is truly random and order is not important, how many possible combinations of lottery numbers are there?
1
2
3
4
5
2345
345
![Page 42: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/42.jpg)
Fundamental Rule Ex: A lottery game selects 3 numbers between 1 and 5
where numbers can not be selected more than once. If the game is truly random and order is not important, how many possible combinations of lottery numbers are there?
1
2
3
4
5
2345
345
LN = 5•4•3
= 60
![Page 43: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/43.jpg)
Combinations
Suppose we flip a coin 3 times, how many ways are there to get 2 heads?
![Page 44: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/44.jpg)
Combinations
Suppose we flip a coin 3 times, how many ways are there to get 2 heads?
Soln:List all possibilities:
H,H,H H,T,TH,H,T H,T,HH,T,H T,H,HT,H,H T,T,T
![Page 45: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/45.jpg)
Combinations
Of 8 possible outcomes, 3 meet criteria
H,H,H H,T,TH,H,T H,T,HH,T,H T,H,HT,H,H T,T,T
![Page 46: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/46.jpg)
Combinations
If we don’t care in which order these 3 occur
H,H,TH,T,HT,H,H
Then we can count by combination.
3 2
3
2 3 2
3 2 1
2 1 13C
!
!( )! ( )
![Page 47: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/47.jpg)
Combinations
Combinations nCk = the number of ways to count k items out n total items order not important.
n = total number of itemsk = number of items pertaining to event A
k nCn
k n k
!
!( )!
![Page 48: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/48.jpg)
Example
How many ways can we select a 4 person committee from 10 students available?
![Page 49: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/49.jpg)
Example
How many ways can we select a 4 person committee from 10 students available?
No. Possible Committees =
10 4
10!
4 6!
10 9 8 7 6!
4 3 2 1 6!1 260C
!
,
![Page 50: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/50.jpg)
Example
We have 20 students, 8 of whom are female and 12 of whom are male. How many committees of 5 students can be formed if we require 2 female and 3 male?
![Page 51: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/51.jpg)
Example
We have 20 students, 8 of whom are female and 12 of whom are male. How many committees of 5 students can be formed if we require 2 female and 3 male?
Soln: Compute how many 2 member female committees we can have and how many 3 member male committees. Each female committee can be combined with each male committee.
![Page 52: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/52.jpg)
Example
8 2 12 3
8!
2 6!
12
3 96 160C C
!
!
! !,
![Page 53: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/53.jpg)
Permutations
Permutations is somewhat like combinations except that order is important.
n kPn
n k
!
( )!
![Page 54: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/54.jpg)
Example
How many ways can a four member committee be formed from 10 students if the first is President, second selected is Vice President, 3rd is secretary and 4th is treasurer?
![Page 55: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/55.jpg)
Example
How many ways can a four member committee be formed from 10 students if the first is President, second selected is Vice President, 3rd is secretary and 4th is treasurer?
10 4
10!
10 45 040P
( )!,
![Page 56: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/56.jpg)
Example
How many ways can a four member committee be formed from 10 students if the first is President, second selected is Vice President, 3rd is secretary and 4th is treasurer?
••
10P4 = 10*9*8*7 = 5,040
![Page 57: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/57.jpg)
![Page 58: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/58.jpg)
South Dakota
School of Mines & Technology
Introduction to Introduction to Probability & StatisticsProbability & Statistics
Industrial Engineering
![Page 59: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/59.jpg)
Introduction to Probability & Statistics
Random VariablesRandom Variables
![Page 60: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/60.jpg)
Random Variables
A Random Variable is a function that associates a real number with each element in a sample space.
Ex: Toss of a die
X = # dots on top face of die = 1, 2, 3, 4, 5, 6
![Page 61: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/61.jpg)
Random Variables
A Random Variable is a function that associates a real number with each element in a sample space.
Ex: Flip of a coin
0 , headsX =
1 , tails
![Page 62: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/62.jpg)
Random Variables
A Random Variable is a function that associates a real number with each element in a sample space.
Ex: Flip 3 coins
0 if TTTX = 1 if HTT, THT, TTH
2 if HHT, HTH, THH 3 if HHH
![Page 63: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/63.jpg)
Random Variables
A Random Variable is a function that associates a real number with each element in a sample space.
Ex: X = lifetime of a light bulb
X = [0, )
![Page 64: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/64.jpg)
Distributions
Let X = number of dots on top face of a die when thrown
p(x) = Prob{X=x}
x 1 2 3 4 5 6
p(x) 1/6 1/6
1/6 1/6
1/6 1/6
![Page 65: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/65.jpg)
Cumulative
Let F(x) = Pr{X < x}
x 1 2 3 4 5 6
p(x) 1/6 1/6
1/6 1/6
1/6 1/6
F(x) 1/6 2/6
3/6 4/6
5/6 6/6
![Page 66: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/66.jpg)
Complementary Cumulative
Let F(x) = 1 - F(x) = Pr{X > x}
x 1 2 3 4 5 6
p(x) 1/6 1/6
1/6 1/6
1/6 1/6
F(x) 1/6 2/6
3/6 4/6
5/6 6/6
F(x) 5/6 4/6
3/6 2/6
1/6 0/6
![Page 67: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/67.jpg)
Discrete Univariate
Binomial Discrete Uniform (Die)
Hypergeometric Poisson Bernoulli Geometric Negative Binomial
![Page 68: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/68.jpg)
Binomial
What is the probability of getting 2 heads out of 3 flips of a coin?
![Page 69: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/69.jpg)
Binomial
What is the probability of getting 2 heads out of 3 flips of a coin?
Soln:H,H,H H,T,TH,H,T T,H,TH,T,H T,T,HT,H,H T,T,T
![Page 70: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/70.jpg)
Binomial
P{2 heads in 3 flips} = P{H,H,T} + P{H,T,H} + P{T,H,H}
= 3•P{H}P{H}P{T}
= 3C2•P{H}2•P{T}3-2
= 3C2•p2•(1-p)3-2
![Page 71: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/71.jpg)
Distributions
Binomial:X = number of successes in n bernoulli trialsp = Pr(success) = const. from trial to trialn = number of trials
p(x) = b(x; n,p) =
n
x n xp px n x!
!( )!( )
1
![Page 72: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/72.jpg)
Binomial Distribution
0.0
0.1
0.2
0.3
0.4
0.5
0 1 2 3 4 5
x
P(x
)
0.0
0.1
0.2
0.3
0.4
0.5
0 1 2 3 4 5
x
P(x
)
n=5, p=.3 n=8, p=.5
x
0.0
0.1
0.2
0.3
0.4
0.5
0 1 2 3 4 5 6 7 8
P(x
)
n=4, p=.8
0.0
0.1
0.2
0.3
0.4
0.5
0 2 4
x
P(x
)
n=20, p=.5
![Page 73: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/73.jpg)
Example
Suppose we manufacture circuit boards with 95% reliability. If approximately 5 circuit boards in 100 are defective, what is the probability that a lot of 10 circuit boards has one or more defects?
![Page 74: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/74.jpg)
Example (soln.)
Pr{ } Pr{ }X X 1 1 0
110
0 1005 95)0 10!
!( !)(. ) (.
= 1 - .9510
= .4013
![Page 75: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/75.jpg)
Example
For p n Pr{X > 1}
.05 10 0.4013
.05 100 0.9941
.05 1,000 1.0000
.01 10 0.0956
.01 100 0.6340
.01 1,000 1.0000
![Page 76: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/76.jpg)
99% Defect Free Rate 500 incorrect surgical procedures every week 20,000 prescriptions filled incorrectly each year 12 babies given to the wrong parents each day 16,000 pieces of mail lost each hour 2 million documents lost by IRS each year 22,000 checks deducted from wrong accounts
during next hour
(Ref: Quality, March 91)
![Page 77: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/77.jpg)
Continuous Distribution
xa b c d
f(x)
A
1. f(x) > 0 , all x
2.
3. P(A) = Pr{a < x < b} =
4. Pr{X=a} =
f x dxa
d
( ) 1
f x dxb
c
( )f x dx
a
a
( ) 0
![Page 78: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/78.jpg)
Continuous Univariate
Normal Uniform Exponential Weibull LogNormal
Beta T-distribution Chi-square F-distribution Maxwell Raleigh Triangular Generalized Gamma H-function
![Page 79: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/79.jpg)
Normal Distribution
65%
95%
99.7%
f x eX
( )
FHG
IKJ1
2
1
2
2
![Page 80: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/80.jpg)
Scale Parameter
x
> 1
= 1
![Page 81: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/81.jpg)
Location Parameter
x
x
> 1
= 1
![Page 82: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/82.jpg)
Std. Normal Transformation
Standard Normal
ZX
f(z)
N(0,1)f z e
z( )
1
2
1
22
![Page 83: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/83.jpg)
Example
Suppose a resistor has specifications of 100 + 10 ohms. R = actual resistance of a resistor and R N(100,5). What is the probability a resistor taken at random is out of spec?
x
LSL USL
100
![Page 84: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/84.jpg)
Example Cont.
x
LSL USL
100
Pr{in spec} = Pr{90 < x < 110}
Pr
90 100
5
110 100
5
x
= Pr(-2 < z < 2)
![Page 85: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/85.jpg)
Example Cont.
x
LSL USL
100
Pr{in spec}= Pr(-2 < z < 2)
= [F(2) - F(-2)]
= (.9773 - .0228) = .9545
Pr{out of spec} = 1 - Pr{in spec}= 1 - .9545= 0.0455
![Page 86: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/86.jpg)
Example
Assume that the per capita income in South Dakota is normally distributed with a mean of $20,000 and a standard deviation of $4,000. If the poverty level is considered to be $15,000 per year, compute the percentage of South Dakotans who would be considered to be at or below the poverty level.
![Page 87: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/87.jpg)
Example
Pr{poverty level} = Pr{X < 15,000}
= Pr{Z < -1.25}
= 0.5 - Pr{0 < Z < 1.25}
= 0.5 - 0.3944 = 0.1056
x
15,000 20,000
}000,4
000,20000,15Pr{
X
![Page 88: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/88.jpg)
Other Continuous Distributions
![Page 89: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/89.jpg)
Exponential Distribution
f x e x( ) Density
Cumulative
Mean 1/
Variance 1/2
F x e x( ) 1
, x > 0
0.0
0.5
1.0
0 0.5 1 1.5 2 2.5 3
Time to Fail
Den
sity
=1
![Page 90: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/90.jpg)
Exponential Distribution
f x e x( ) Density
Cumulative
Mean 1/
Variance 1/2
F x e x( ) 1
, x > 0
=1
0.0
0.5
1.0
1.5
2.0
0 0.5 1 1.5 2 2.5 3
Time to Fail
Den
sity =2
![Page 91: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/91.jpg)
0.0
0.5
1.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
x
f(x)
LogNormal
Density
Cumulative no closed form
Mean
Variance
f xx
ex
( )ln
1
2
1
2
2
, x > 0
e 2 2
e e2 2 2
1 ( ) = 0
=1
![Page 92: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/92.jpg)
0.0
0.5
1.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
x
f(x)
LogNormal
Density
Cumulative no closed form
Mean
Variance
f xx
ex
( )ln
1
2
1
2
2
, x > 0
e 2 2
e e2 2 2
1 ( ) = 0
=2
![Page 93: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/93.jpg)
-0.5
0.0
0.5
1.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
x
f(x)
LogNormal
Density
Cumulative no closed form
Mean
Variance
f xx
ex
( )ln
1
2
1
2
2
, x > 0
e 2 2
e e2 2 2
1 ( ) = 0
=0.5
![Page 94: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/94.jpg)
Gamma
Density
Cumulative no closed form for integer
Mean
Variance 2
f x x e x( )( )
/
1 , x > 0
0.0
0.5
1.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
x
f(x) =1
![Page 95: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/95.jpg)
0.0
0.5
1.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
x
f(x)
Gamma
Density
Cumulative no closed form for integer
Mean
Variance 2
f x x e x( )( )
/
1 , x > 0
=2
![Page 96: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/96.jpg)
0.0
0.5
1.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
x
f(x)
Gamma
Density
Cumulative no closed form for integer
Mean
Variance 2
f x x e x( )( )
/
1 , x > 0
=3
![Page 97: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/97.jpg)
0.0
0.5
1.0
1.5
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
Weibull
Density
Cumulative
Mean Variance
f x x e x( ) ( / ) 2 1 2
, x > 0
F x e x( ) ( / ) 12
1
2 2
22 1 1
= 1
= 1
![Page 98: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/98.jpg)
0.0
0.5
1.0
1.5
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
Weibull
Density
Cumulative
Mean Variance
f x x e x( ) ( / ) 2 1 2
, x > 0
F x e x( ) ( / ) 12
1
2 2
22 1 1
= 1
= 2
![Page 99: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/99.jpg)
0.0
0.5
1.0
1.5
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
Weibull
Density
Cumulative
Mean Variance
f x x e x( ) ( / ) 2 1 2
, x > 0
F x e x( ) ( / ) 12
1
2 2
22 1 1
= 1
= 3
![Page 100: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/100.jpg)
Uniform
Density
Cumulative
Mean (a + b)/2
Variance (b - a)2/12
f xb a
( )1
, a < x < b
F xx a
b a( )
f(x)
x
a b
![Page 101: Introduction to Probability & Statistics South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering](https://reader033.vdocuments.mx/reader033/viewer/2022061609/56649ce55503460f949b2a0e/html5/thumbnails/101.jpg)
End
Probability Review Session 1