Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:1 Environmentally Conscious Design & Manufacturing Class 25: Probability

Environmentally Conscious Design & Manufacturing (ME592)

Date: May 5, 2000 Slide:1

Environmentally Conscious Design & Manufacturing

Class 25: Probability and Statistics

Prof. S. M. Pandit



Agenda

• Random Variable

• Mean and variance

• Normal distribution

• Sampling

• Linear regression



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.



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

;



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]:



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



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



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



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(--



Normal Distribution

-3

68.27%

95.45%

99.73%

-2 - + +2 +3



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(



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(



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



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



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



Data Example: Grinding Wheel Profile



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”



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



Simple Linear Regression



Normal Distribution of yt

Observation=Prediction+Error

tt

tt1t

εy

εxβy

ˆ



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




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




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



Regression Equation with Observed Data



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

Documents

Environmentally Conscious Design & Manufacturing (ME592) Date: May 5, 2000 Slide:1 Environmentally Conscious Design & Manufacturing Class 25: Probability