Download Environmentally Conscious Design & Manufacturing ME592E-1

Environmentally Conscious
Design & Manufacturing
Class 25: Probability and Statistics
Prof. S. M. Pandit
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000
Slide:1
Agenda
• Random Variable
• Mean and variance
• Normal distribution
• Sampling
• Linear regression
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000
Slide:2
Random Variables
Random variables are numerical-valued quantities
whose observed values are governed by the laws of
probability.
•
•
Discrete random variables: the random variable X can
take on only one of several discrete values x1, x2,…,
xn and no other value.
Continuous random variable: the random variable X
can take on a nondenumerably infinite number of
values.
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000
Slide:3
Continuous Random Variables
The cumulative distribution function F(x):
F(x)  P(X  x)
0  F(x)  1
F(x1 )  F(x 2 ) if x1  x 2 ;
lim F(x)  1
x 
lim F(x)  0
x  
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000
Slide:4
Continuous Random Variables
The probability density function f(x):
d
f(x) 
F(x)
dx
The probability of occurrence of interval [a,b]:
b
P(a  X  b)   f(x)dx F(b) - F(a)
a
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000
Slide:5
Moments of Random Variables

The moments of the distribution
mr 
r
x
 f(x)dx

The first -order moment is called the mean, the expected
value or the expectation E[X]:
E X  

 xf(x)dx  μ
x

2
The second -order moment is called the variance σ x :

σ x2   (x -μ x ) 2 f(x)dx  E(X - μ x ) 2  0

Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000
Slide:6
Properties of Expectation
E(cX)=cE(X),
where c is a constant
E(X+Y)=E(X)+E(Y)
E(XY)=E(X)E(Y)
if X & Y are independent
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000
Slide:7
Theorems on Variance
Var(cX)=c2Var(X)
Var(X+Y)=Var(X)+Var(Y) (s2x+y=s2x+s2y)
if X and Y are independent
Var(X-Y)=Var(X)+Var(Y) (s2x-y=s2x+s2y)
if X and Y are independent
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000
Slide:8
Normal Distribution
Probability density

1
f(x) 
e
2π
(x  μ x ) 2
2σ x 2
,σ x  0,  μ x  
b
P(a  X  b)   f(x)dx F(b) - F(a)
a

P(-   X   )   f(x)dx 1
-
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000
Slide:9
Normal Distribution
99.73%
95.45%
68.27%
-3
-2
-

+
+2
+3
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000
Slide:10
Standard Normal Distribution
Let Z 
X  μx
σx
1
μ z  E(Z) 
(μ x  μ x )  0
σx
Z ~ N(0,1)
σ z2  E(Z  μ z )2  E(Z 2 )  1
Cumulative probabilities
Φ(z)  P(   z )
Φ(1.65)  0.95053 Φ(1.96)  0.975 Φ(2.58)  0.99506
P( 1.65  Z  1.65)  Φ(1.65)  Φ( 1.65)
 Φ(1.65)  (1  Φ( 1.65))  90%
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000
Slide:11
Standard Normal Distribution
P( 1.65  Z  1.65)  Φ(1.65)  Φ( 1.65)
 Φ(1.65)  (1  Φ( 1.65))  90%
P( 1.96  Z  1.96)  Φ(1.96)  Φ( 1.96)
 Φ(1.96)  (1  Φ( 1.96))  95%
P( 2.58  Z  2.58)  Φ(2.58)  Φ( 2.58)
 Φ(2.58)  (1  Φ( 2.58))  99%
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000
Slide:12
Sampling
Population and sample
Population

N
μ x  E(X), σ x2  E(X  μ x )2
X
Sample average
Sample N
N
1
Xi

N i 1
1 N
x   xi
N i 1
Sample variance
1 n
2
S 
(x

x
)
 i
N  1 i 1
2
d
Sd is the sample
standard deviation
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000
Slide:13
Estimator of Sample Mean & Variance
E(X)  E(

1
1
1
1
X
)

E(
X

X

...

XN )

i
1
2
N
N
N
N
1
1
1
μ
μ
μ
E(X 1 )  E(X 2 )  ...  E(X N )  X  X  ...  X  μ X
N
N
N
N
N
N
1
1
1
X 1  X 2  ...  X N )
N
N
N
1
1
1
NVar(X) Var(X)
 2 Var(X 1 )  2 Var(X 2 )  ...  2 Var(X N ) 

2
N
N
N
N
N
Var(X)  Var(
1
σ  E(X  E(X))  σ X2
N
2
X
2
σx 
1
σx
N
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000
Slide:14
Confidence Interval
X  μ x  Zσ x  mean  Z * standard deviation
90%, 95%, 99% probability limits on X are
μ x  1.65σ x , μx  1.96σ x , μ x  2.58σ x
σ x2
X ~ N(μ x , )
N
(1-) % confidence interval on the mean μ x is
x  zα/2 σ x / N
where
Φ(zα/2 )  1 - α/2
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000
Slide:15
Data Example: Grinding Wheel Profile
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000
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Linear Regression
To express the dependence of one set of observations yt
on another set xt under the assumption that yt’s are
independent or uncorrelated.
model
y t  β0  β1 x t  εt , t  1,2,..., N
N
“best fit”
βˆ 0  y  βˆ1 x and βˆ1 
 (y
t 1
t
 y)(x t  x)
N
2

(
x

x
)
 t
t 1
where
1 N
1 N
x   x t , y   y t
N i 1
N i 1
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000
Slide:17
Least Squares Estimates
To minimize the sum of squares of the “ residuals” t’s
Let
y t  y  y t , x t  x  xt
y t  β1 x t  ε t , t  1,2,..., N, εt ~ NID(0,σ ε2 )
NID-Normally Independently Distributed
N
β̂1 
y x
t 1
N
t
 xt
2
t
N
1
residual sum of squares
2
ˆ
ˆ
σ   (y t  β1 x t ) 
N t 1
Number of residuals
2
ε
t 1
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000
Slide:18
Simple Linear Regression
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000
Slide:19
Normal Distribution of yt
Observation=Prediction+Error
y t  β1 x t  ε t
 yˆ t  ε t
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000
Slide:20
Computations Examples
t  1 2 3 4 5 N  5 
x t  5 6 3 2 5
y t  7 6 5 4 6
1
1

x

 t 5 (5  6  3  2  5)  4.2
N t
1
1
y   y t  (7  6  5  4  6)  5.6
N t
5
x
Removing the mean yields
t 1
2
3
4 5
x t  0.8 1.8 - 1.2 - 2.2 0.8
y t  1.4 0.4 - 0.6 - 1.6 0.4
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000
Slide:21
Computations Examples
N
βˆ1 
y x
t 1
N
t
 xt
2
t

6.4
 0.59
10.8
t 1
5
1
σˆ   (y t  0.59x t ) 2
5 t 1
1
 [(0.928 2  ( 0.662) 2  (0.108) 2  ( 0.302) 2  ( 0.072) 2  0.281
5
2
ε
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000
Slide:22
Computations Examples
Assuming that these estimated values are true values
y t  β1 x t  ε t
 0.59x t  ε t
ε t ~ NID(0,0.281)
ε t  y t  0.59x t
yˆ t  β1 x t  0.59x t
The % 95 probability limits for the observation yt are
yˆ t  1.96σε  β1 xt  1.96σε  0.59x t  1.96  0.281  0.59x t  1.04
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000
Slide:23
Regression Equation with Observed Data
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000
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Homework #8
The problems 1 through 2 are out of the textbook “Industrial Ecology”
1. Problem 14.2 (Answer: Φ  37.8GJ/t, Ω  0.77, Ψ  0.23 )
2. Problem 14.3 (Answer: Φ  43.8075GJ/ t, Ω  0.9625, Ψ  0.23 )
3. List some of the new environmentally friendly energy technologies.
4. Discuss and illustrate the contrast between the traditional and loss
function based approaches to characterize quality.
5. Discuss and illustrate the shortcomings of the loss function
approach and how they can be overcome by the satisfaction metric
that includes benefits.
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000
Slide:25