Chapter IX · 26 min

Continuous Chance

Density, not mass — and which curve

“The error to which our observations are subject follows a law which can be represented by a curve.”
— After Carl Friedrich Gauss, Theoria motus corporum coelestium (1809)

Not every chance lives on a list. A waiting time, a measurement error, a model’s predicted probability — these sit on a continuum. The new object is a density: a function whose integrals, not values, are probabilities. The value $f (x)$ is not $P (X = x)$ . That last probability is zero.

Density is not mass

A random variable is continuous (absolutely continuous, if one is being careful) when there is a non-negative integrable function $f_{X}$ such that

P (a \leq X \leq b) = \int_{a}^{b} f_{X} (x) d x, \int_{- \infty}^{\infty} f_{X} = 1.

The cdf is still $F_{X} (x) = P (X \leq x)$ , and now $F_{X}^{'} (x) = f_{X} (x)$ at continuity points. Intervals of the same length in a tall part of the density are more probable than those in a low part — that is all the picture says.

Word	Symbol	What it is	How to use it
pmf	p(k)	a mass at a point, for discrete X	sum p(k) over a set
density	f(x)	a height; not a probability	integrate f over an interval
cdf	F(x)	P(X ≤ x), always	P(a < X ≤ b) = F(b) − F(a)
support	where f > 0	the values that can actually happen	uniform lives on [a,b]; exponential on [0, ∞)

A field guide

Family	Story	Lives on	Mean
Uniform	equally likely on an interval	[a, b]	(a + b) / 2
Exponential	wait with no memory	[0, ∞)	1/λ
Normal	sum of many small independent bits	all of ℝ	μ (spread σ)

Three more names you will meet later, and can ignore until you need them: Beta lives on $(0, 1)$ and is a prior on a coin’s $p$ ; Gamma generalises the exponential to waiting for several ticks; Student-t is a normal with a heavier tail, used when the variance is estimated from a small sample. They are not required for the rest of this book.

Uniform and exponential

The uniform law on $[a, b]$ has density $1/ (b - a)$ there and zero elsewhere. It is the continuous analogue of “equally likely”: equal-length subintervals are equally likely. The standard uniform $U \sim Unif (0, 1)$ is the raw material of simulation: any cdf that can be inverted yields a random variable as $F^{- 1} (U)$ .

Uniform density on [0, 1]. Probability is length. The shaded interval has chance equal to its width.

The exponential law with rate $λ$ has density $λ e^{- λ x}$ on $[0, \infty)$ , mean $1/ λ$ , and the same memorylessness as the geometric: given survival to time $t$ , the remaining wait is a fresh exponential. It is the continuous waiting time for a Poisson process to tick once.

Exponential density with rate 1. The shaded early interval carries more chance than a later interval of the same length.

The normal law

The normal (Gaussian) family $N (μ, σ^{2})$ has density

f (x) = \frac{1}{σ 2 π} exp (- \frac{( x - μ ) ^{2}}{2 σ ^{2}}) .

Mean $μ$ , variance $σ^{2}$ . The standard normal $Z \sim N (0, 1)$ is the case $μ = 0, σ = 1$ ; any other is a scaled shift, $X = μ + σ Z$ . To turn a question about $X$ into a question about $Z$ , write

P (X \leq x) = P (Z \leq \frac{x - μ}{σ}) .

That fraction is the only linear algebra this family needs: subtract the centre, divide by the spread. The empirical rule: about 68% of the mass sits within one standard deviation of the mean, 95% within two, 99.7% within three. These are integrals of this curve, not commandments of nature — but many measurements are close enough that the numbers are useful.

The standard normal density. The teal region is P(Z ≤ 0.8). Area, not height, is chance. Open the bell lab to move the fences.

Why this curve, among all bell-shaped functions? Because sums of many small independent contributions, suitably scaled, become normal. That is the central limit theorem, Chapter 11. It is also why Gaussian noise is the default in so many models: not because the world is Gaussian, but because leftover error is often a sum.

Try the bell lab.