Probability (Ch. 6)

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. average standard deviation sample size

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

4. Normal Distributions Let 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 (0X/2).

6. 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.

7. Student’s t Distribution

8. Chapter 7Uncertainty Analysis

9. Plot X-Y data with uncertainties 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 • 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

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 + 35.678360310.2 1270.178 383710.2 first “doubtful” digits “doubtful” digits 1270.2 383700

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 _____.) 1000

14. 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

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 =± [ ] = (0.02)*(3.65) + (0.01) ±0.083 V ±[0.073 + 0.01] = 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 = ±0.1% of reading + 4 digits 4 digits in the least significant place First “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 = ±0.5% of reading + 20 digits First “doubtful” digit - ending zeros to the right of decimal points ARE significant!

19. 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

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

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

22. 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?

23. Blank Page (Notes on board)

24. 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)

25. Uncertainty Analysis #5 • Carrying out the partial derivatives,

26. Uncertainty Analysis #6 • Substituting in the nominal values,

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

28. Uncertainty Analysis #12 • Simplified approach:

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:

31. Exercise #1c • Equation:

32. Exercise #1d • Answers:

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

34. Exercise #2a • Show that

35. Exercise #2b • Base form • Simplified form

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

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