Mathematics

Probability, continued

Expected value, variance, normal density, and a CLT one-liner.

Basics

Expected value

Discrete: E[X] = Σ x P(X=x). Continuous: E[X] = ∫ x f(x) dx. It is the probabilistic mean. Linearity E[aX+b] = a E[X] + b always holds. For independent X,Y, E[XY] = E[X] E[Y]. Some laws (Cauchy) have no expectation. The sample mean estimates E[X].

Variance

Var(X) = E[(X − μ)²] = E[X²] − μ². It measures spread in squared units; σ = √Var returns to the original unit. Var(aX+b) = a² Var(X). For independent summands, variances add. Chebyshev bounds tails without knowing the shape. Outliers swing variance hard.

Normal density

The normal (Gaussian) density is f(x) = (1/√(2πσ²)) exp(−(x−μ)²/(2σ²)). μ centres it; σ sets the width. It is bell-shaped with mean = median = mode. Standardise Z=(X−μ)/σ to use tables. About 68%, 95%, 99.7% lie in ±1σ, ±2σ, ±3σ. Many measurement-error models assume this curve.

CLT one-liner

If draws are i.i.d. with finite variance, the distribution of the sample mean grows closer to normal as n grows. Even a skewed parent yields a bell for the mean when n is large. That is the classroom one-liner for the central limit theorem. Fine print and rates belong in a probability course. Many intervals and tests lean on this approximation.

Formulas

Expectation

E[X] = Σ x p(x) or ∫ x f(x) dx

Discrete sum or continuous integral.

Symbols

p(x) probability mass
f(x) probability density

Variance

Var(X) = E[X²] − (E[X])², σ = √Var(X)

σ is the standard deviation.

Symbols

μ E[X]

Normal density

f(x) = 1/√(2πσ²) · exp(−(x−μ)²/(2σ²))

μ mean, σ>0 standard deviation.

Symbols

μ mean
σ standard deviation

Sample mean (CLT sketch)

X̄ ≈ Normal(μ, σ²/n) (n large)

Variance of the mean shrinks as σ²/n.

Symbols

n sample size

Key table

Linearity	E[X+Y]=E[X]+E[Y] even without independence
68–95–99.7	rough fractions inside μ±σ, ±2σ, ±3σ for a normal
Cauchy	no mean or variance; a foil for CLT’s finite-variance premise