1

Probability (Ch. 6)

►Probability: “…the chance of occurrence of an event in an experiment.” [Wheeler & Ganji]

►Chance: “…3. The probability of anything happening; possibility.” [Funk & Wagnalls]

A measure of how certain we are that a particular outcome will occur.

2

Probability Distribution Functions

►Descriptors of the distribution of data. Require some parameters:

► _______, _______________. Degrees of freedom (__________) may be

required for small sample sizes. Called “probability density functions” for

continuous data.►Typical distribution functions:

Normal (Gaussian), Student’s t.

averagestandard deviation

sample size

3

Probability Density Functions dxxxxP ii dxxf i

Suggests integration!

bxaP b

a

i dxxf

-5 0 50

0.1

0.2

0.3

0.4

x

f(x) 22 2/xe2

1xf

Normal Normal Probability Probability

Density Function:Density Function:

=0=1

4

Normal Distributions

-5 0 50

0.1

0.2

0.3

0.4

x

f(x) 22 2/xe

2

1xf

Let zx

2/z2e2

1zf

bxaP ,zzzP 21 1z 2za

b

Transform your data to zero-mean, =1, and evaluate probabilities in that domain!

5

Normal Distribution► Standard table available describing the area under the curve

from “0 to z” for a normal distribution. (Table 6.3 from Wheeler and Ganji.) So, if you want X%, look for (0X/2).

6

Student’s t DistributionData with nData with n30.30.

Based on calculating the Based on calculating the area of the shaded area of the shaded

portions.portions.Total area = Total area =

t/2-t/2

2/2/ txtP 1

Result we’re looking Result we’re looking for:for:

n

Stx 2/

1w/ confidence:w/ confidence:

How do we get How do we get tt/2/2??

7

Student’s t Distribution

88

Chapter 7Uncertainty Analysis

9

Plot X-Y data with uncertainties

Time Voltage Uncert.1 35 1.51.5 23 1.022 17 0.783 12 0.584 9 0.465 7 0.386 6 0.348 4 0.2610 3 0.22

Where do these come from?

10

Significant Digits

►In ME 360, we will follow the rules for significant digits

►Be especially careful with computer generated output

►Tables created with Microsoft Excel are particularly prone to having…

- excessive significant digits!

11

Rules for Significant Digits

leastleast

233.5^2 =233.5^2 =

►In multiplication, division, and other In multiplication, division, and other operations, carry the result to the same operations, carry the result to the same number of significant digits that are in the number of significant digits that are in the quantity used in the equation with the _____ quantity used in the equation with the _____ number of significant digits.number of significant digits.

234^2 =234^2 =

If we expand the limits of If we expand the limits of uncertainty:uncertainty:

54756 --> 54756 -->

5480054800

234.5^2 =234.5^2 =

54522.25 --54522.25 -->>54990.25 --54990.25 -->>

5452054520

5499054990

12

Rules for Significant Digits

► In addition and subtraction, do not carry the result past the ____ column containing a doubtful digit (going left to right).

1234.5 23400

+ 35.678 360310.2

1270.178 383710.2 1270.21270.2 383700383700

firstfirst

““doubtfuldoubtful” digits” digits

““doubtfuldoubtful” digits” digits

13

Rules for Significant Digits► In a lengthy computation, carry extra significant

digits throughout the calculation, then apply the significant digit rules at the end.

►As a general rule, many engineering values can be assumed to have 3 significant digits when no other information is available.

► (Consider: In a decimal system, three digits implies 1 part in _____.)10001000

14

Sources of Uncertainty

1. Precision uncertainty Repeated measurements of same value Typically use the ____ (±2S) interval

2. ___ uncertainty from instrument3. Computed Uncertainty

Technique for determining the uncertainty in a result computed from two or more uncertain values

95%

Bias

15

Instrument Accuracy►Measurement accuracy/uncertainty often

depends on scale setting►Typically specified as

ux = % of reading + n digits Example:

DMM reading is 3.65 V with uncertainty (accuracy) of ±(2% of reading + 1 digit):

ux =± [ ]

=

(0.01)(0.01)(0.02)*(3.65) +(0.02)*(3.65) +

±[0.073 + 0.01] =±[0.073 + 0.01] =±0.083 V±0.083 V

DON’T FORGET!DON’T FORGET!

16

Instrument Accuracy

►Data for LG Precision #DM-441B True RMS Digital Multimeter

►What is the uncertainty in a measurement of 7.845 volts (DC)??

17

DMM (digital multimeter)For DC voltages in the 2-20V range,

accuracy =

V004.0V845.7100

1.0yUncertaint

4 digits in the least significant place

±0.1% of reading + 4 digits

V011845.0

V012.0First “doubtful” digit

18

DMM (digital multimeter)►What is the uncertainty in a

measurement of 7.845 volts AC at 60 Hz? For AC voltages in the 2-20V, 60 Hz range,

accuracy =

V020.0V845.7100

5.0yUncertaint

±0.5% of reading + 20 digits

V059225.0V059.0

First “doubtful” digit - ending zeros to the right of decimal points ARE significant!

19

Sources of Uncertainty

1. Precision uncertainty Repeated measurements of same value Typically use the ____ (±2S) interval

2. ___ uncertainty from instrument3. Computed Uncertainty

Technique for determining the uncertainty in a result computed from two or more uncertain values

95%

Bias

20

►We want to experimentally determine the uncertainty for a quantity W, which is calculated from 3 measurements (X, Y, Z)

2

31

4Z

YXW

Uncertainty Analysis #1

231 ZYX4

21

►The three measurements (X, Y, Z) have nominal values and bias uncertainty estimates of

N/m0.203.70X m0.051.36Y N0.102.30Z

Uncertainty Analysis #2

22

►The nominal value of the quantity W is easily calculated from the nominal measurements,

►What is the uncertainty, uW in this value for W?

W

Uncertainty Analysis #3

N

m38.1

2

2

31

N)4(2.30

m)(1.36N/m)70.3(

23

Blank Page (Notes on board)

24

► To estimate the uncertainty of quantities computed from equations:

► Note the assumptions and restrictions given on p. 182! (Independence of variables, identical confidence levels of parameters)

2

Z

2

Y

2

X

W

u

Z

W

W

u

Y

W

W

u

X

W

Uncertainty Analysis #4

W

uW2

z

2

y

2

x uZ

Wu

Y

Wu

X

W

W

1

25

X

W

Y

W

Z

W

Uncertainty Analysis #5►Carrying out the partial derivatives,

23ZY4

221 ZYX4

3

331 ZYX4

2

231 ZYX4

W

2

3

N

m373.0

N

m05.3

2

2

N

m202.1

26

N

m38.1

N

m373.0

W

1

X

W2

2

3

m

112.2

N

m38.1

N

m202.1

W

1

Z

W2

2

2

Uncertainty Analysis #6

►Substituting in the nominal values,

N

m72.0

N

m38.1

N

m05.3

W

1

Y

W2

N

0.87

27

W

u

X

W x

m

112.2

W

u

Z

W z

Uncertainty Analysis #7

►Substituting in the nominal values,

N

m72.0

W

u

Y

W y

N

0.87

m

N2.0 054.0

m05.0 1105.0

N1.0 087.0

Square the Square the terms, sum, terms, sum, and get the and get the square-root:square-root:

0.02269525

7479830.15064942

%1.15

28

►Simplified approach:

Uncertainty Analysis #12

W

uW

2312

31

ZYX44Z

YXW

W

uW

2

Z

2

Y

2

X

Z

u2

Y

u3

X

u1

22

2

N3.2

N10.02

m36.1

m05.03

m

N3.7

m

N0.201 151.0

29

Uncertainty Analysis #14►Which of the three measurements X,

Y, or Z, contribute the most to the uncertainty in W?

►If you wanted to reduce your uncertainty in the measured W, what should you do first?

30

Exercise #1a

►Experimental gain from an op-amp circuit is found from the formula

►Compute the uncertainty in gain, uG, if both Ein and Eout have uncertainty:

in

out

E

EG

volts08.065.2E in volts11.027.6E out

1in

1outEE

31

G

uG

Exercise #1c►Equation: 11 inout

in

out EEE

EG

2

in

E

2

out

E

E

u1

E

u1 inout

32

►Answers:

Exercise #1d

in

out

E

EG

G

uG

volt65.2

volt27.637.2

Gu038.0

Gu37.2038.0

%8.3

Gu 09.0

22

volt65.2

volt08.01

volt27.6

volt11.01

33

Exercise #2

3ML

EI3

►What is the uncertainty in if E, M, and L are all uncertain?

U

23212121 LMIE3

2

L

2

M

2

E U

L

U

M

U

E

34

Exercise #2a

►Show that

1

E E2

1

1

M M2

1

1

L L2

3

35

Exercise #2b►Base form

► Simplified form

U

2

L

2

M

2

E

L

U

2

3

M

U

2

1

E

U

2

1

2

L

2

M

2

E U

L

U

M

U

E

U

36

►Compute the nominal value for and the uncertainty with these values:

in

seclbf-2.04.2M

2

in5.11.25L

Exercise #2c

2in

lbf5E106E2.10E

4in012.0I

37

►Use Eqn. 7.11 (p. 165)

►generally compute intermediate uncertainties at the 95% confidence level

Combining Bias and Precision Uncertainties

22 yuncertaintprecisionyuncertaintbiasw