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.

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. average standard deviation sample size

Probability Density Functions Suggests integration! Normal Probability Density Function: =0 =1

Normal Distributions Let Transform your data to zero-mean, =1, and evaluate probabilities in that domain!

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).

Student’s t Distribution Data with n30. Result we’re 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.

Student’s t Distribution

Chapter 7 Uncertainty Analysis Student Pages: 1,2,3,4,5,6 8,9,10,12,13,14 15,16,17,18,20,23 In PDF file leave out 6, 10

Plot X-Y data with uncertainties Where do these come from?

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!

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 234^2 = 54756 --> 54800 If we expand the limits of uncertainty: 233.5^2 = 54522.25 --> 54520 234.5^2 = 54990.25 --> 54990

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

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

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

ux = % of reading + n digits 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 =± [ ] = (0.02)*(3.65) + (0.01) ±[0.073 + 0.01] = ±0.083 V DON’T FORGET!

Instrument Accuracy Data for LG Precision #DM-441B True RMS Digital Multimeter What is the uncertainty in a measurement of 7.845 volts (DC)?? pass out multimeter of each - see Barry or Dan for analog type?

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

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!

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

Uncertainty Analysis #1 We want to experimentally determine the uncertainty for a quantity W, which is calculated from 3 measurements (X, Y, Z)

Uncertainty Analysis #2 The three measurements (X, Y, Z) have nominal values and bias uncertainty estimates of

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?

Blank Page (Notes on board)

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)

Uncertainty Analysis #5 Carrying out the partial derivatives,

Uncertainty Analysis #6 Substituting in the nominal values,

Uncertainty Analysis #7 Substituting in the nominal values, Square the terms, sum, and get the square-root:

Uncertainty Analysis #12 Simplified approach:

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?

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:

Exercise #1c Equation:

Exercise #1d Answers:

Exercise #2 What is the uncertainty in w if E, M, and L are all uncertain?

Exercise #2a Show that

Exercise #2b Base form Simplified form

Exercise #2c Compute the nominal value for w and the uncertainty with these values:

Combining Bias and Precision Uncertainties Use Eqn. 7.11 (p. 165) generally compute intermediate uncertainties at the 95% confidence level