1 probability (ch. 6) ► probability: “…the chance of occurrence of an event in an...
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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?
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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!
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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
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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
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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
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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!
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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
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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
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►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
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►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(
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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
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►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
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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
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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
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