Probability (Ch. 6)

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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.
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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), Students t.

average
standard deviation
sample size
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Probability Density Functions
Suggests integration!
Normal Probability Density Function
?0 ?1
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Normal Distributions
Let
?Transform your data to zero-mean, ?1, and
evaluate probabilities in that domain!
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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 (0?X/2).

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Students t Distribution
Data with n?30.
Result were looking for
a/2
a/2
w/ confidence
ta/2
-ta/2
How do we get ta/2?
Based on calculating the area of the shaded
portions. Total area a.
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Students t Distribution
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Chapter 7Uncertainty Analysis
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Plot X-Y data with uncertainties
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

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

least
2342
54756 --gt
54800
If we expand the limits of uncertainty
233.52
54522.25 --gt
54520
234.52
54990.25 --gt
54990
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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

first
doubtful digits
doubtful digits
1270.2
383700
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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 _____.)

1000
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Sources of Uncertainty

• Precision uncertainty
• Repeated measurements of same value
• Typically use the ____ (2S) interval
• ___ uncertainty from instrument
• Computed Uncertainty
• Technique for determining the uncertainty in a
  result computed from two or more uncertain values

95
Bias
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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)
(0.02)(3.65)
0.083 V
0.073 0.01
DONT 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)??

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DMM (digital multimeter)

• For DC voltages in the 2-20V range, accuracy

0.1 of reading 4 digits
4 digits in the least significant place
First 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

0.5 of reading 20 digits
First doubtful digit - ending zeros to the
right of decimal points ARE significant!
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Sources of Uncertainty

• Precision uncertainty
• Repeated measurements of same value
• Typically use the ____ (2S) interval
• ___ uncertainty from instrument
• Computed Uncertainty
• Technique for determining the uncertainty in a
  result computed from two or more uncertain values

95
Bias
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Uncertainty Analysis 1

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

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Uncertainty Analysis 2

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

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Uncertainty Analysis 3

• The nominal value of the quantity W is easily
  calculated from the nominal measurements,
• What is the uncertainty, uW in this value for W?

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Blank Page (Notes on board)
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Uncertainty Analysis 4

• 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)

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Uncertainty Analysis 5

• Carrying out the partial derivatives,

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Uncertainty Analysis 6

• Substituting in the nominal values,

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Uncertainty Analysis 7

• Substituting in the nominal values,

Square the terms, sum, and get the square-root
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Uncertainty Analysis 12

• Simplified approach

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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?

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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

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Exercise 1c

• Equation

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Exercise 1d

• Answers

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Exercise 2

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

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Exercise 2a

• Show that

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Exercise 2b

• Base form
• Simplified form

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Exercise 2c

• Compute the nominal value for w and the
  uncertainty with these values

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Combining Bias and Precision Uncertainties

• Use Eqn. 7.11 (p. 165)
• generally compute intermediate uncertainties at
  the 95 confidence level