Mathematics and Statistics Flashcards

The Fourier series of a periodic function requires which condition on the function over its period?

Answer: The function must satisfy Dirichlet conditions (bounded, finite extrema, finite discontinuities)

Dirichlet conditions — bounded variation, finite number of maxima/minima and discontinuities — are sufficient for Fourier series convergence.

For a binomial distribution with n = 10 trials and p = 0.3, the mean number of successes is:

Answer: 3.0

The mean of a binomial distribution is μ = np = 10 × 0.3 = 3.0.

What is the rank of the matrix [[1,2,3],[2,4,6],[0,0,0]]?

Answer: 1

Row 2 is twice row 1, and row 3 is zero, so only one row is linearly independent, giving rank = 1.

The trapezoidal rule for numerical integration approximates the area under a curve as a series of:

Answer: Trapezoids connecting consecutive data points

The trapezoidal rule connects consecutive function values with straight lines, forming trapezoids whose areas are summed.

The gradient of f(x,y) = x²y + 3y² evaluated at (1, 2) is:

Answer: (4, 13)

∂f/∂x = 2xy = 4, ∂f/∂y = x² + 6y = 1 + 12 = 13, so ∇f(1,2) = (4, 13).

A chi-square goodness-of-fit test is used to determine whether:

Answer: Observed frequencies match expected frequencies from a theoretical distribution

The chi-square test compares observed counts to expected counts to assess how well data fits a hypothesized distribution.

What is the solution to the system of equations: 2x + y = 5 and x - y = 1?

Answer: x = 2, y = 1

Adding the equations: 3x = 6, so x = 2; substituting back: y = 5 - 2(2) = 1.