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Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000 Slide:1
Environmentally Conscious Design & Manufacturing
Class 25: Probability and Statistics
Prof. S. M. Pandit
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000 Slide:2
Agenda
• Random Variable
• Mean and variance
• Normal distribution
• Sampling
• Linear regression
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000 Slide:3
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:4
Continuous Random Variables
The cumulative distribution function F(x):
x)P(XF(x) 1F(x)0
0F(x)lim
1F(x)lim
xx if )F(x)F(x
x
x
2121
;
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000 Slide:5
Continuous Random Variables
The probability density function f(x):
F(x)dx
df(x)
F(a)-F(b)f(x)dxb)XP(ab
a
The probability of occurrence of interval [a,b]:
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000 Slide:6
Moments of Random Variables
f(x)dxxm rrThe moments of the distribution
The first -order moment is called the mean, the expected value or the expectation E[X]:
xμxf(x)dxXE
The second -order moment is called the variance :
0)μ-E(Xf(x)dx)-μ(xσ 2x
2x
2x
2xσ
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000 Slide:7
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:8
Theorems on Variance
Var(cX)=c2Var(X)
Var(X+Y)=Var(X)+Var(Y) (s2x+y=s2
x+s2y)
if X and Y are independent
Var(X-Y)=Var(X)+Var(Y) (s2x-y=s2
x+s2y)
if X and Y are independent
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000 Slide:9
Normal Distribution
xx2σ
)μ(x
μ0,σ,e2π
1f(x)
2x
2x
Probability density
F(a)-F(b)f(x)dxb)XP(ab
a
1f(x)dx)XP(--
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000 Slide:10
Normal Distribution
-3
68.27%
95.45%
99.73%
-2 - + +2 +3
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000 Slide:11
Standard Normal Distribution
Cumulative probabilities
x
x
σ
μXZ
0)μ(μσ
1E(Z)μ xx
xz
1)E(Z)μE(Zσ 22z
2z N(0,1)~Z
)zP(Φ(z)
Let
0.95053Φ(1.65) 0.975Φ(1.96) 0.99506Φ(2.58)
90%1.65))Φ((1Φ(1.65)
1.65)Φ(Φ(1.65)1.65)Z1.65P(
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000 Slide:12
Standard Normal Distribution
90%1.65))Φ((1Φ(1.65)
1.65)Φ(Φ(1.65)1.65)Z1.65P(
95%1.96))Φ((1Φ(1.96)
1.96)Φ(Φ(1.96)1.96)Z1.96P(
99%2.58))Φ((1Φ(2.58)
2.58)Φ(Φ(2.58)2.58)Z2.58P(
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000 Slide:13
Population and sample PopulationN
Population and sample Population
Sample N
2x
2xx )μE(XσE(X),μ
N
1iiX
N
1X
n
1i
2i
2d )x(x
1N
1S
Sample average
Sample varianceSd is the sample standard deviation
Sampling
N
1iix
N
1x
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000 Slide:14
Estimator of Sample Mean & Variance
XXXX
N21
N21i
μN
μ...
N
μ
N
μ)E(X
N
1...)E(X
N
1)E(X
N
1
)XN
1...X
N
1X
N
1E()X
N
1E()XE(
N
Var(X)
N
NVar(X))Var(X
N
1...)Var(X
N
1)Var(X
N
1
)XN
1...X
N
1X
N
1Var()XVar(
2N22212
N21
2X
22X
σN
1))XE(XE(σ xx σ
N
1σ
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000 Slide:15
Confidence Interval
deviationstandard*ZmeanZσμX xx
90%, 95%, 99% probability limits on X are
xxxxxx 2.58σμ1.96σμ1.65σμ ,,
)~N
σ,N(μX
2x
x
xμ (1-) % confidence interval on the mean is
N/σzx xα/2
where α/2-1)Φ(zα/2
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000 Slide:16
Data Example: Grinding Wheel Profile
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000 Slide:17
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.
N1,2,...,t,εxββy tt10t
where
model
N
1t
2t
N
1ttt
110
)xx(
)xx)(yy(βandxβyβ
ˆˆˆ
N
1it
N
1it y
N
1y,x
N
1x
“best fit”
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000 Slide:18
Least Squares Estimates
NID-Normally Independently Distributed
tttt xxx,yyy
)σNID(0,~εN,1,2,...,t,εxβy 2εttt1t
To minimize the sum of squares of the “ residuals” t’sLet
N
1t
2t
N
1ttt
1
x
xyβ̂
residualsofNumber
squaresofsumresidual)xβ(y
N
1σ
N
1t
2t1t
2ε
ˆˆ
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000 Slide:19
Simple Linear Regression
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000 Slide:20
Normal Distribution of yt
Observation=Prediction+Error
tt
tt1t
εy
εxβy
ˆ
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000 Slide:21
Computations Examples
5.66)456(75
1y
N
1y
4.25)236(55
1x
N
1x
64567y
52365x
5N54321t
tt
tt
t
t
0.41.6-0.6-0.41.4y
0.82.2-1.2-1.80.8x
54321t
t
t
Removing the mean yields
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000 Slide:22
Computations Examples
0.5910.8
6.4
x
xyβ N
1t
2t
N
1ttt
1
ˆ
0.2810.072)(0.302)((0.108)0.662)([(0.9285
1
)0.59x(y5
1σ
22222
5
1t
2tt
2ε
ˆ
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000 Slide:23
Computations Examples
Assuming that these estimated values are true values
tt1t
ttt
t
tt
tt1t
0.59xxβy
0.59xyε
1)NID(0,0.28~ε
ε0.59x
εxβy
ˆ
The % 95 probability limits for the observation yt are
1.040.59x0.2811.960.59x1.96σxβ1.96σy ttεt1εt ˆ
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000 Slide:24
Regression Equation with Observed Data
Environmentally Conscious Design & Manufacturing (ME592)
Date: May 5, 2000 Slide:25
Homework #8
The problems 1 through 2 are out of the textbook “Industrial Ecology”
1. Problem 14.2 (Answer: )
2. Problem 14.3 (Answer: )
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.
0.23Ψ0.77,Ω37.8GJ/t,Φ 0.23Ψ0.9625,Ωt,43.8075GJ/Φ