Advanced Mathematics and Statistics Reference

Posted on Feb 7, 2026 in Mathematics

Probability Density Functions and Distributions

A function f(x) is a valid probability density function (PDF) if:

• f(x) ≥ 0 for all values of x.
• ∫_-∞^∞ f(x)dx = 1.

Expectation and Variance

• Expected value (mean): E[X] = ∫_-∞^∞ x ⋅ f(x)dx
• Expected value of X²: E[X²] = ∫_-∞^∞ x² ⋅ f(x)dx
• Variance: Var(X) = E[X²] – (E[X])²

Special Probability Models

• Uniform Distribution: f(x) = 1/(b-a), for a ≤ x ≤ b.
• Uniform Distribution Expected Value: (a + b)/2
• Exponential Distribution: f(x) = λe^-λx, for x ≥ 0
• Standard Normal Distribution: P(a ≤ X ≤ b) = Φ(b) – Φ(a)

Taylor and Maclaurin Series

General Form of a Taylor Polynomial

P_n(x) = f(a) + f'(a)(x – a) + [f”(a)/2!](x-a)² + [f”'(a)/3!](x-a)³ + … + [f⁽ⁿ⁾(a)/n!](x – a)ⁿ

Maclaurin Series Formula

f(x) = f(0) + f'(0)x + [f”(0)/2!](x²) + [f”'(0)/3!](x³) + … + [f⁽ⁿ⁾(0)/n!](x)ⁿ + …

Solving a Maclaurin Polynomial

1. Compute the first four derivatives.
2. Evaluate each derivative at x = 0.
3. Plug the results into the Maclaurin series formula.

Multivariable Calculus and Optimization

Partial Derivatives

Example: If f(x,y) = x^2 + y^2, then the partial derivatives are ∂f/∂x = 2x and ∂f/∂y = 2y.

Critical Points and Classification

1. To find critical points, find first-order partial derivatives and set them to 0.
2. To classify the critical point as a local maximum, minimum, or saddle point, compute the second partial derivatives (f_xx, f_yy, f_xy).
3. Define the discriminant: D = f_xxf_yy – (f_xy)².

• If D > 0 and f_xx > 0: Local minimum.
• If D > 0 and f_xx < 0: Local maximum.
• If D < 0: Saddle point.
• If D = 0: Inconclusive.

Lagrange Multipliers

Used to maximize or minimize a function f(x, y) subject to the constraint g(x, y) = c.

1. Set up the gradient equation: ∇f = λ∇g, which implies f_x = λg_x and f_y = λg_y.
2. Compute ∇f and ∇g.
3. Solve the three equations (including the constraint g(x, y) = c) for x, y, and λ.
4. Plug x and y into f(x, y) to find the maximum or minimum values.

Integration Techniques and Applications

Area and Volume

• Area between two curves (vertical): ∫_a^b [f(x) – g(x)] dx, where f(x) is the top function and g(x) is the bottom function over the interval [a, b].
• Volume of Revolution (Disk Method): V = π ∫_a^b [R(x)]² dx, where R(x) is the outer radius.
• Volume of Revolution (Washer Method): V = π ∫_a^b ([R(x)]² – [r(x)]²) dx, where R(x) is the outer radius and r(x) is the inner radius.

Integration Methods

• Integration by Parts: ∫ u dv = uv – ∫ v du. Choose u as the function to differentiate and dv as the function to integrate.
• Double Integrals: ∫₀² ∫₀¹ (x + y) dx dy. Integrate the inner integral first, then the outer integral.
• Average Value of a Function: (1 / (b – a)) ∫_a^b f(x) dx.

Basic Trigonometric Integrals

• ∫ sin x dx = -cos x + C
• ∫ cos x dx = sin x + C
• ∫ sec²x dx = tan x + C
• ∫ csc²x dx = -cot x + C
• ∫ sec x tan x dx = sec x + C
• ∫ csc x cot x dx = -csc x + C

Improper Integrals

• Converges: If the limit approaches a finite value.
• Diverges: If the limit approaches infinity or does not exist.

Sequences, Series, and Financial Math

Geometric Sequences and Series

• Geometric Sequence n^th term: a_n = a₁(r)^n-1, where r is the common ratio.
• Convergence: Converges to 0 if |r| < 1.
• Partial Sum Formula: S_n = a₁(1 – rⁿ) / (1 – r) if r ≠ 1.
• Infinite Series Sum: S_∞ = a₁ / (1 – r), valid only if |r| < 1.
• Geometric Mean: √(a ⋅ b)

Money Flow and Annuities

• Total Money Flow: ∫_a^b R(t) dt. Used for total income or inflow generated from a stream.
• Present Value of Money Flow: ∫_a^b R(t) ⋅ e^-rt dt, where R(t) is the rate of income and e^-rt is the discount factor.
• Future Value of an Annuity: P ⋅ [(1 + r)ⁿ – 1] / r
• Present Value of an Annuity: P ⋅ [1 – (1 + r)^-n] / r

Differential Equations (ODE)

General Forms

• General ODE: dy/dx = f(x, y)
• Separable ODE: dy/dx = g(x)h(y)
• Linear ODE: dy/dx + P(x)y = Q(x)
• Exponential Growth/Decay: y(t) = y₀e^kt
• Radioactive Decay: dN/dt = -kN → N(t) = N₀e^-kt, where k > 0 is the decay constant.

Euler’s Method

For first-order ODEs in the form dy/dx = f(x, y) with initial condition y(x₀) = y₀:

• Euler Formula: y_n+1 = y_n + h ⋅ f(x_n, y_n)
• Steps: 1) Identify the differential equation, initial value, step size h, and number of steps. 2) Apply the formula iteratively.

Predator-Prey Model (Lotka-Volterra)

• Prey: dx/dt = ax – bxy
• Predator: dy/dt = -cy + dxy
• Variables: x(t) is the number of prey, y(t) is the number of predators, a is the prey birth rate, b is the predation rate, c is the predator death rate, and d is the predator growth rate.