Mathematics and Statistics Flashcards

The curl of vector field F = (y, -x, 0) at any point is:

Answer: (0, 0, -2)

curl F = (∂Fz/∂y - ∂Fy/∂z, ∂Fx/∂z - ∂Fz/∂x, ∂Fy/∂x - ∂Fx/∂y) = (0, 0, -1-1) = (0, 0, -2).

What is the coefficient of variation (CV) for a dataset with mean = 80 and standard deviation = 16?

Answer: 20%

CV = (σ/μ) × 100% = (16/80) × 100% = 20%.

Which iterative method for solving linear systems Ax = b splits A into lower and upper triangular parts and updates each variable using the most recent values?

Answer: Gauss-Seidel method

The Gauss-Seidel method uses each updated variable immediately within the same iteration, unlike Jacobi which uses only previous-iteration values.

What is the inverse Laplace transform of 1/(s² + 4)?

Answer: (1/2)sin(2t)

L⁻¹{ω/(s²+ω²)} = sin(ωt), so L⁻¹{1/(s²+4)} = (1/2)sin(2t) since ω = 2.

In linear regression y = mx + b fitted to data, the slope m minimizes which quantity?

Answer: Sum of squared residuals

Ordinary least squares regression finds the slope and intercept that minimize the sum of squared residuals (SSR).

The unit vector in the direction of A = (3, 4) is:

Answer: (3/5, 4/5)

|A| = √(9+16) = 5, so the unit vector is (3/5, 4/5) = (0.6, 0.8).

A fair die is rolled twice. What is the probability of getting a sum of 7?

Answer: 1/6

There are 6 favorable outcomes {(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)} out of 36 total, so P = 6/36 = 1/6.