South%20Dakota%20%20School%20of%20Mines%20

1
South Dakota School of Mines Technology
Introduction to Probability Statistics
2
Exam III

• 99
• 97
• 95
• 93
• 90
• 87
• 86
• 85
• 83
• 82,82
• 80
• Average 88.25

3
Introduction to Probability StatisticsRandom
Variables
4
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

5
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 , heads
• X
• 1 , tails

?
6
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 TTT
• X 1 if HTT, THT, TTH
• 2 if HHT, HTH, THH
• 3 if HHH

?
7
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, ?)

8
Distributions

• Let X number of dots on top face of a die
  when thrown
• p(x) ProbXx

9
Cumulative

• Let F(x) PrX lt x

10
Complementary Cumulative

• Let F(x) 1 - F(x) PrX gt x

11
Discrete Univariate

• Binomial
• Discrete Uniform (Die)
• Hypergeometric
• Poisson
• Bernoulli
• Geometric
• Negative Binomial

12
Binomial Distribution
n5, p.3
n8, p.5
n20, p.5
n4, p.8
x
13
Binomial Measures

• Mean
• Variance

?
?
?
xp
x
(
)
np
x
np(1-p)
14
Continuous Distribution
f(x)
A
x
a b c
d

• 1. f(x) gt 0 , all x
• 2.
• 3. P(A) Pra lt x lt b
• 4. PrXa

15
Continuous Univariate

• Beta
• T-distribution
• Chi-square
• F-distribution
• Maxwell
• Raleigh
• Triangular
• Generalized Gamma
• H-function

• Normal
• Uniform
• Exponential
• Weibull
• LogNormal

16
Normal Distribution
65
95
99.7
17
Std. Normal Transformation
f(z)
Standard Normal
N(0,1)
18
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?

19
Example Cont.
Prin spec Pr90 lt x lt 110
Pr(-2 lt z lt 2)
20
Example Cont.
Prin spec
Pr(-2 lt z lt 2) F(2) - F(-2) (.9773 -
.0228) .9545
21
Example Cont.
Prin spec
Pr(-2 lt z lt 2) F(2) - F(-2) (.9773 -
.0228) .9545
Prout of spec 1 - Prin spec 1 -
.9545 0.0455
22
Exponential Distribution
?1
23
Exponential Distribution
?1
?2
24
Example

• Let X lifetime of a machine where the life is
  governed by the exponential distribution.
  determine the probability that the machine fails
  within a given time period a.
• , x gt 0, ? gt 0

25
Example
Exponential Life
2.0
1.8
1.6
1.4
1.2
f(x)
Density
1.0
0.8
0.6
0.4
0.2
0.0
0
0.5
1
1.5
2
2.5
3
a
Time to Fail
26
Example
Exponential Life
2.0
1.8
1.6
1.4
1.2
f(x)
Density
1.0
0.8
0.6
0.4
0.2
0.0
0
0.5
1
1.5
2
2.5
3
a
Time to Fail
27
Example
Note F(?) 1-e-?? 1 F(0) 1 - e-?0 0
28
Complementary
Exponential Life

• Suppose we wish to know the probability that the
  machine will last at least a hrs?

2.0
1.8
1.6
1.4
1.2
f(x)
Density
1.0
0.8
0.6
0.4
0.2
0.0
0
0.5
1
1.5
2
2.5
3
a
Time to Fail
?
29
Example

• Suppose for the same exponential distribution, we
  know the probability that the machine will last
  at least a more hrs given that it has already
  lasted c hrs.

a
c
ca
PrX gt a c X gt c PrX gt a c ? X gt c /
PrX gt c PrX gt a c / PrX gt c
30
Introduction to Probability StatisticsExpecta
tions
31
Expectations

• Mean

32
Example
Consider the discrete uniform die example

• ?? EX 1(1/6) 2(1/6) 3(1/6)
• 4(1/6) 5(1/6) 6(1/6)
• 3.5

33
Variance
34
Example

• Consider the discrete uniform die example

?2 E(X-?)2 (1-3.5)2(1/6) (2-3.5)2(1/6)
(3-3.5)2(1/6) (4-3.5)2(1/6) (5-3.5)2(1/6)
(6-3.5)2(1/6) 2.92
35
Property
36
Property
37
Property
38
Example

• Consider the discrete uniform die example

?2 EX2 - ?2 12(1/6) 22(1/6) 32(1/6)
42(1/6) 52(1/6) 62(1/6) - 3.52
91/6 - 3.52 2.92
39
Exponential Example
For a product governed by an exponential life
distribution, the expected life of the product
is given by
2.0
1.8
1.6
1.4
1.2
?
x
?
f
t )
e
(x
?
?
1.0
Density
0.8
0.6
0.4
0.2
X
0.0
0
0.5
1
1.5
2
2.5
3
0.5
1
1/?
40
Properties of Expectations

• 1. Ec c
• 2. EaX b aEX b
• 3. ?2(ax b) a2?2
• 4. Eg(x)
• g(x) Eg(x)
• X
• (x-?)2
• e-tx

41
Properties of Expectations

• 1. Ec c
• 2. EaX b aEX b
• 3. ?2(ax b) a2?2
• 4. Eg(x)
• g(x) Eg(x)
• X ?
• (x-?)2 ?2
• e-tx ?(t)

42
Class Problem

• Total monthly production costs for a casting
  foundry are given by
• TC 100,000 50X
• where X is the number of castings made during a
  particular month. Past data indicates that X is
  a random variable which is governed by the normal
  distribution with mean 10,000 and variance 500.
  What is the distribution governing Total Cost?

43
Class Problem

• Soln
• TC 100,000 50X
• is a linear transformation on a normal
• TC Normal(mTC, s2TC)

44
Class Problem

• Using property Eaxb aExb
• mTC E100,000 50X
• 100,000 50EX
• 100,000 50(10,000)
• 600,000

45
Class Problem

• Using property s2(axb) a2s2(x)
• s2TC s2(100,000 50X)
• 502 s2(X)
• 502 (500)
• 1,250,000

46
Class Problem

• TC 100,000 50 X
• but,
• X N(100,000 , 500)
• TC N(600,000 , 1,250,000)
• N(600000 , 1118)

47
Introduction to Probability StatisticsJoint
Expectations
48
Properties of Expectations

• Recall that
• 1. Ec c
• 2. EaX b aEX b
• 3. ?2(ax b) a2?2
• 4. Eg(x)

49
Joint Expectation

• Let Z X Y
• EZ EX EY

50
Joint Expectation

• Let Z X Y
• EZ EX EY
• For X, Y independent, cov(x,y) 0

51
In General

• In general, for Z the sum of n independent
  variables, Z X1 X2 X3 . . . Xn

52
Class Problem

• Suppose n customers enter a store. The ith
  customer spends some Xi amount. Past data
  indicates that Xi is a uniform random variable
  with mean 100 and standard deviation 5.
  Determine the mean and variance for the total
  revenue for 5 customers.

53
Class Problem

• Total Revenue is given by
• TR X1 X2 X3 X4 X5

54
Class Problem

• Total Revenue is given by
• TR X1 X2 X3 X4 X5
• Using the property EZ EX EY,
• ETR EX1 EX2 EX3 EX4 EX5

55
Class Problem

• Total Revenue is given by
• TR X1 X2 X3 X4 X5
• Using the property EZ EX EY,
• ETR EX1 EX2 EX3 EX4 EX5
• 100 100 100 100 100
• 500

56
Class Problem

• TR X1 X2 X3 X4 X5
• Use the property s2(Z) s2(X) s2(Y).
• Assuming Xi, i 1, 2, 3, 4, 5 all independent
• s2(TR) s2(X1) s2(X2) s2(X3) s2(X4)
  s2(X5)

57
Class Problem

• TR X1 X2 X3 X4 X5
• Use the property s2(Z) s2(X) s2(Y).
• Assuming Xi, i 1, 2, 3, 4, 5 all independent
• s2(TR) s2(X1) s2(X2) s2(X3) s2(X4)
  s2(X5)

58
Class Problem

• TR X1 X2 X3 X4 X5
• Use the property s2(Z) s2(X) s2(Y).
• Assuming Xi, i 1, 2, 3, 4, 5 all independent
• s2(TR) s2(X1) s2(X2) s2(X3) s2(X4)
  s2(X5)
• 52 52 52 52 52
• 125

59
Class Problem

• TR X1 X2 X3 X4 X5
• Xi , independent with mean 100 and standard
  deviation 5
• ETR 500
• s2(TR) 125