Continuous Probability

Grinstead & Snell · GFDL · PDF

When the sample space is a continuum, individual points have probability zero. Probability lives in areas under curves. A density function f(x) replaces the discrete weight function, and P(a < X < b) = ∫ f(x) dx from a to b.

From counting to measuring

Spin a wheel, pick a random real number between 0 and 1, measure a person's height. The sample space is an interval (or all of R). You cannot list the outcomes, so you cannot assign a positive probability to each one. Instead, you describe how probability is distributed across regions.

Scheme

; Density function: probability = area under the curve
; For uniform on [0,1]: f(x) = 1 everywhere

; Approximate integral by summing rectangles (Riemann sum)
(define (integrate f a b n)
  (let* ((dx (/ (- b a) n))
         (sum (let loop ((i 0) (s 0))
                (if (>= i n) s
                    (loop (+ i 1) (+ s (* (f (+ a (* i dx))) dx)))))))
    sum))

; Uniform density on [0,1]
(define (uniform-density x)
  (if (and (>= x 0) (<= x 1)) 1 0))

(display "P(0 < X < 1)   = ")
(display (integrate uniform-density 0 1 1000)) (newline)
(display "P(0 < X < 0.5) = ")
(display (integrate uniform-density 0 0.5 1000)) (newline)
(display "P(0.3 < X < 0.7) = ")
(display (integrate uniform-density 0.3 0.7 1000))

Density functions

A density function f(x) satisfies two rules (the continuous versions of the discrete axioms):

f(x) ≥ 0 for all x
∫ f(x) dx over the entire real line = 1

The density at a point is not a probability. It is a rate: how quickly probability accumulates near that point. Only areas (integrals) give probabilities.

Scheme

; A triangular density on [0, 2]: f(x) = x for 0<=x<=1, f(x) = 2-x for 1<x<=2
(define (triangle-density x)
  (cond ((and (>= x 0) (<= x 1)) x)
        ((and (> x 1) (<= x 2)) (- 2 x))
        (else 0)))

; Verify it integrates to 1
(display "total area = ")
(display (integrate triangle-density 0 2 1000)) (newline)

; P(0.5 < X < 1.5)
(display "P(0.5 < X < 1.5) = ")
(display (integrate triangle-density 0.5 1.5 1000))

The normal distribution

The most important continuous distribution. Its density is the bell curve: f(x) = (1/√(2π)) e^(-x²/2) for the standard normal (mean 0, variance 1). About 68% of the area lies within one standard deviation of the mean, 95% within two.

Scheme

; Standard normal density: f(x) = (1/sqrt(2*pi)) * e^(-x^2/2)
(define pi 3.14159265358979)
(define e  2.71828182845905)

(define (normal-density x)
  (* (/ 1 (sqrt (* 2 pi)))
     (expt e (* -0.5 (* x x)))))

; P(-1 < X < 1) should be about 0.6827
(display "P(-1 < X < 1) = ")
(display (integrate normal-density -1 1 2000)) (newline)

; P(-2 < X < 2) should be about 0.9545
(display "P(-2 < X < 2) = ")
(display (integrate normal-density -2 2 2000)) (newline)

; P(-3 < X < 3) should be about 0.9973
(display "P(-3 < X < 3) = ")
(display (integrate normal-density -3 3 2000))

Cumulative distribution function

The CDF is F(x) = P(X ≤ x) = ∫ f(t) dt from -∞ to x. It rises from 0 to 1 and is non-decreasing. The density is the derivative of the CDF: f(x) = F'(x). Two ways to describe the same distribution.

Scheme

; CDF: integrate density from -infinity to x
; For uniform on [0,1]: F(x) = x for 0<=x<=1

(define (uniform-cdf x)
  (cond ((< x 0) 0)
        ((> x 1) 1)
        (else x)))

(display "F(0.0) = ") (display (uniform-cdf 0.0)) (newline)
(display "F(0.3) = ") (display (uniform-cdf 0.3)) (newline)
(display "F(0.7) = ") (display (uniform-cdf 0.7)) (newline)
(display "F(1.0) = ") (display (uniform-cdf 1.0)) (newline)

; P(0.3 < X < 0.7) = F(0.7) - F(0.3)
(display "P(0.3 < X < 0.7) = ")
(display (- (uniform-cdf 0.7) (uniform-cdf 0.3)))

Notation reference

Textbook	Scheme	Meaning
f(x)	(normal-density x)	Density function
F(x) = P(X ≤ x)	(uniform-cdf x)	Cumulative distribution function
∫ f(x)dx = 1	(integrate f a b n)	Total area = 1 (normalization)
P(a < X < b)	(integrate f a b n)	Probability = area under curve
N(0, 1)	(normal-density x)	Standard normal distribution

Neighbors

Adjacent chapters

🎰 Ch 1: Discrete Probability — the finite version of everything on this page
🎰 Ch 3: Combinatorics — counting, which powers discrete probability
🎰 Ch 4: Conditional Probability — conditioning works for continuous distributions too

Paper pages

🍞 Fritz 2020 — Markov categories handle both discrete and continuous probability in one framework
🍞 Staton 2025 — probabilistic programs sample from continuous distributions
🍞 Baez-Fritz 2011 — entropy as a functor, connecting information to probability

Foundations (Wikipedia)

Ready for the real thing? Read Ch 2 of Grinstead & Snell.

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