Mathematics and Statistics Flashcards

The eigenvalues of the matrix [[4,1],[2,3]] are:

Answer: 5 and 2

The characteristic equation is (4-λ)(3-λ) - 2 = λ² - 7λ + 10 = 0, giving λ = 5 and λ = 2.

If f(x) = x³ - 6x² + 9x, at which x-value does a local minimum occur?

Answer: x = 3

f'(x) = 3x² - 12x + 9 = 0 gives x = 1 (local max) and x = 3 (local min), verified by the second derivative test.

In hypothesis testing, a Type I error is defined as:

Answer: Rejecting a true null hypothesis

A Type I error (false positive) occurs when a true null hypothesis is incorrectly rejected; its probability equals α.

What is the value of the integral ∫₀^π sin(x) dx?

Answer: 2

∫₀^π sin(x) dx = [-cos(x)]₀^π = -cos(π) + cos(0) = 1 + 1 = 2.

The dot product of vectors A = (3, 4) and B = (1, -2) is:

Answer: -5

A · B = (3)(1) + (4)(-2) = 3 - 8 = -5.

Which of the following describes a Poisson distribution?

Answer: Number of events in a fixed time interval given a known average rate

The Poisson distribution models the number of rare, independent events occurring in a fixed interval at a constant mean rate λ.

Using Simpson's 1/3 rule with n=2 intervals to approximate ∫₀^2 x² dx (h=1), the result is:

Answer: 2.67

Simpson's rule gives (h/3)[f(0) + 4f(1) + f(2)] = (1/3)[0 + 4 + 4] = 8/3 ≈ 2.67, matching the exact answer.