What the characteristic function φ(t) = E[e^(itX)] does: Fourier transform of the PDF

Write the formula for normal distribution PDF

Normal distribution PDF: 𝑋/(σ√2π) * e^(-(X-μ)^2/(2σ^2))

What the Cayley-Hamilton theorem says: every matrix satisfies its own characteristic equation

Cayley-Hamilton theorem: A square matrix satisfies its characteristic polynomial

What the Boltzmann distribution describes — probability of a state with energy E is proportional to e^(-E/kT)

Boltzmann distribution: Probability of state E ∝ e^(-E/kT)

Write the formula for Pearson correlation coefficient

r = Σ((xi - x̄)(yi - ȳ)) / (√Σ(xi - x̄)² * √Σ(yi - ȳ)²)

Write the equation for sigmoid function σ(x) = 1/(1+e^-x)

σ(x) = 1 / (1 + e^-x)

Write the formula for covariance between X and Y

Cov(X, Y) = Σ((Xi - X̄)(Yi - Ȳ)) / (n - 1)