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Propagation of uncertainty - Quiz
Contributed by: Craven
  • 1. What happens to the variance of the difference A - B when A and B are uncorrelated?
A) The variance will be more than either A or B.
B) The variance will decrease.
C) The variance will become zero.
D) The variance will remain unchanged.
  • 2. What is the effect on the variance of A - B when A and B are perfectly positively correlated?
A) The variance becomes twice that of either A or B.
B) The variance converges to zero.
C) The variance remains the same as when uncorrelated.
D) The variance increases.
  • 3. What happens to the variance of A - A if A is perfectly autocorrelated?
A) It quadruples the input variance.
B) It has zero variance.
C) It remains unchanged.
D) It doubles the input variance.
  • 4. If A is uncorrelated, what is the output variance for f = A - A?
A) Zero variance.
B) Twice the input variance.
C) The same as the input variance.
D) Four times the input variance.
  • 5. What happens to the output variance of f = A - A if A is perfectly anticorrelated?
A) It quadruples the input variance.
B) It becomes zero.
C) It halves the input variance.
D) It remains unchanged.
  • 6. What is the derivative of f(x) = arctan(x)?
A) df/dx = 1/x
B) df/dx = x/(1 + x2)
C) df/dx = 1/(1 + x2)
D) df/dx = -1/(1 + x2)
  • 7. How is the propagated uncertainty Δf for f(x) = arctan(x) calculated?
A) Δf ≈ Δx / (1 + x2)
B) Δf ≈ Δx - (1 + x2)
C) Δf ≈ Δx * (1 + x2)
D) Δf ≈ Δx + (1 + x2)
  • 8. In resistance measurement using Ohm's law, what is the formula for uncertainty σR?
A) σR ≈ R * sqrt((σV/V)2 + (σI/I)2)
B) σR ≈ σV * σI
C) σR ≈ σV / σI
D) σR ≈ V / I
  • 9. What assumption is made about the correlation between current and voltage in the resistance measurement example?
A) Their possible correlation is neglected.
B) They are assumed to be perfectly anticorrelated.
C) Their correlation is considered variable.
D) They are assumed to be perfectly correlated.
  • 10. What happens to the variance of A - B as the positive correlation between A and B increases?
A) The variance increases.
B) The variance becomes zero immediately.
C) The variance remains unchanged.
D) The variance decreases.
  • 11. If A is perfectly anticorrelated, what happens to the input variance in f = aA - aA?
A) It is halved.
B) It is quadrupled in the output.
C) It remains unchanged.
D) It becomes zero.
  • 12. What is the propagated uncertainty for f(x) = arctan(x) when x = 0?
A) Δf ≈ Δx
B) Δf ≈ 0
C) Δf ≈ Δx2
D) Δf ≈ 1/Δx
  • 13. What is the effect of a negative correlation on the variance of A - B?
A) It makes the variance zero.
B) It has no effect.
C) It decreases the variance.
D) It increases the variance compared to the uncorrelated case.
  • 14. What is the uncertainty in resistance R if V and I are perfectly correlated?
A) The formula σR ≈ R * sqrt((σV/V)2 + (σI/I)2) does not account for correlation.
B) The uncertainty doubles.
C) The uncertainty becomes zero.
D) The uncertainty halves.
  • 15. What is the variance of A - B when A and B are perfectly negatively correlated?
A) It decreases.
B) It is increased compared to the uncorrelated case.
C) It becomes zero.
D) It remains unchanged.
  • 16. How does the uncertainty in resistance change if σV increases while keeping σI constant?
A) The uncertainty remains unchanged.
B) The uncertainty in resistance decreases.
C) The uncertainty in resistance increases.
D) The uncertainty becomes zero.
  • 17. What happens to the variance of A - B as the negative correlation between A and B increases?
A) The variance decreases.
B) The variance remains unchanged.
C) The variance becomes zero.
D) The variance further increases.
  • 18. Which statistical measure is commonly used to quantify uncertainty?
A) Median absolute deviation
B) Mean absolute error
C) Standard deviation, σ
D) Range
  • 19. What is the most general way to characterize uncertainty?
A) By specifying its probability distribution
B) Through mean and median values
C) Using only standard deviation
D) By calculating variance only
  • 20. When uncertainties are correlated, what must be taken into account?
A) Covariance
B) Variance alone
C) Absolute error
D) Standard deviation only
  • 21. Which method is commonly used to propagate uncertainty in nonlinear functions?
A) Monte Carlo methods
B) Simple algebraic procedures
C) Frequency distribution analysis
D) Linear regression
  • 22. What is the matrix notation for the linear combination of variables x?
A) 𝐟 = 𝐀𝐱
B) 𝐟 = 𝐱 + 𝐀
C) 𝐟 = 𝐀 ⊗ 𝐱
D) 𝐟 = 𝐱/𝐀
  • 23. In component notation, how is the variance-covariance matrix of f expressed?
A) Σijf = ∑knAikσkxAjk
B) Σijf = AikΣkxAjk
C) Σijf = ∑kn∑lnAikΣklxAjl
D) Σijf = 𝐀𝜮x
  • 24. When the errors on x are uncorrelated, how does the expression for Σijf simplify?
A) Σijf = ∑kn∑lnAikΣklxAjl
B) Σijf = AikσkxAjk
C) Σijf = ∑knAikΣkxAjk
D) Σijf = 𝐀𝜮x
  • 25. What is the expression for σf² when f = ab and variables a and b are uncorrelated?
A) (σff)² ≈ 2abσab
B) (σff)² ≈ σa² + σb²
C) (σff)² ≈ b²σa² + a²σb²
D) (σff)² ≈ (σaa)² + (σbb)²
  • 26. How is the variance of a product ABC derived for uncorrelated variables?
A) (σff)² ≈ (σaa)² + (σbb)² + (σcc)²
B) (σff)² ≈ σa² + σb² + σc²
C) (σff)² ≈ A²B²C²
D) (σff)² ≈ 3(σaa)²
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