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