LESSON 0.1 · ABOUT 20 MINUTES
Calculus & notation readiness
Refresh the tools that probability calculations depend on.
By the end of this lesson
- Interpret sums, derivatives, and definite integrals.
- Use a short diagnostic to choose what to review.
Read the notation
Σ adds terms; ∫ adds density over an interval. P(X ≤ x) is a cumulative probability, not a density. Keep the random variable X separate from a possible value x.
Differentiate and integrate
For n ≠ −1, the antiderivative of xⁿ is xⁿ⁺¹/(n+1). Evaluate a definite integral by subtracting the antiderivative at its lower endpoint from its value at the upper endpoint. The derivative of e^(ax) is a·e^(ax).
Series and probability
For |r| < 1, 1 + r + r² + … = 1/(1−r). This often normalizes a probability mass function. A probability distribution must have nonnegative masses or density, with total probability one.
∫ₐᵇ f(x) dx = F(b) − F(a), where F′ = f
Worked example
A density is f(x)=c x² on 0<x<2. Find c.
Integrate c x² from 0 to 2: c·8/3=1, so c=3/8. The density may exceed one at a point; its total area must equal one.