How to read mathematical statistics notation

A formula usually does not fail to read because the concept is unknown. It fails because the symbols cannot be said out loud. Skim ∫₀^∞ e^(tx) λe^(−λx) dx with your eyes and it passes; say "the integral from zero to infinity" and the structure appears.

So the readings came first. Symbol → how to say it → what it means.

1. Greek letters

Symbol	Say it as	Meaning
α	alpha	Significance level, a parameter
β	beta	A parameter, Type II error probability
γ	gamma	Parameter of the gamma distribution
Γ	capital gamma	The gamma function
λ	lambda	Parameter of the exponential and Poisson distributions
μ	mu	Population mean
σ	sigma	Population standard deviation
σ²	sigma squared	Population variance
θ	theta	An unknown parameter
ρ	rho	Population correlation coefficient
π	pi	The circle constant, or a probability
φ	phi	Standard normal density
Φ	capital phi	Standard normal cumulative distribution function
ε	epsilon	Error term, a very small quantity
δ	delta	A small change, an error

2. Random variables and probability

Symbol	Say it as	Meaning
X	big X	A random variable
x	little x	The value X actually took
Y	big Y	Another random variable
P(X=x)	the probability that X equals x	Discrete probability
P(A)	the probability of A	Probability that event A occurs
E(X)	the expectation of X	The mean
E[g(X)]	the expectation of g of X	Mean of the function g(X)
Var(X)	the variance of X	How much it varies
Cov(X,Y)	the covariance of X and Y	How far two variables move together
Corr(X,Y)	the correlation of X and Y	Linear relationship between two variables

3. Distributions

Symbol	Say it as	Meaning
f(x)	f of x	Probability density or mass function
F(x)	capital F of x	Cumulative distribution function
F_X(x)	capital F sub X of x	The CDF of X
f_X(x)	f sub X of x	The density of X
P(X≤x)	the probability that X is at most x	Definition of the CDF
X ~ N(μ, σ²)	X is distributed normal	Mean μ, variance σ²
X ~ Exp(λ)	X is distributed exponential	Parameter λ
X ~ Bin(n, p)	X is distributed binomial	n trials, success probability p
X ~ Pois(λ)	X is distributed Poisson	Parameter λ
X ~ Beta(α, β)	X is distributed beta	Parameters α, β

~ is read "is distributed". X ~ N(μ, σ²) is "the random variable X is distributed normal with mean μ and variance σ²."

4. Sums, products, integrals, derivatives

Symbol	Say it as	Meaning
Σ	sigma	A sum
Σᵢ₌₁ⁿ	the sum from i equals one to n	Add from i=1 to n
∏	capital pi	A product
∫	integral	Summation over a continuous variable
∫₋∞^∞	the integral from minus infinity to infinity	Over the whole range
∞	infinity	A value growing without bound
∂	partial	Derivative with respect to one of several variables

5. Statistical inference

Symbol	Say it as	Meaning
X̄	X bar	Sample mean
S²	S squared	Sample variance
S	S	Sample standard deviation
θ̂	theta hat	An estimator or estimate of θ
H₀	H naught	Null hypothesis
H₁	H one	Alternative hypothesis
p-value	p value	The p value
α	alpha	Significance level

The fifteen learned first

Fifty at once was not the plan. These fifteen keep coming back while reading the rest.

α alpha · β beta · λ lambda · μ mu · σ sigma · σ² sigma squared · θ theta · Γ the gamma function · f(x) density · F(x) CDF · E(X) expectation · Var(X) variance · Σ sum · ∫ integral · ∞ infinity

Three that kept getting confused

σ and Σ. In Korean both are said "sigma" and the meanings differ. English keeps them slightly apart — sigma and capital sigma — but the shapes are still the reliable difference.

• σ standard deviation
• Σ a sum

Γ and γ

• Γ capital gamma — the gamma function
• γ lowercase gamma — a parameter of the gamma distribution

X and x. This one took longest.

• X the random variable. Nothing has been fixed yet
• x the value that X was actually observed to take

The difference between upper and lower case carries the difference between possibility and result.

A formula actually worked through

The moment generating function of the exponential distribution:

M_X(t) = ∫₀^∞ e^(tx) · λe^(−λx) dx

Said aloud — "M sub X of t is the integral from zero to infinity of e to the t x times lambda e to the minus lambda x, d x."

Simplified:

M_X(t) = λ / (λ − t),    for t < λ

Said aloud — "M sub X of t is lambda over lambda minus t, for t less than lambda."

Once it could be read, the reason for the condition t < λ came first. The integral has to converge, and that follows the moment the symbols are spoken.

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Formulas use Unicode symbols rather than LaTeX. This site has no math renderer, and adding KaTeX would blow the per-route budget on CSS and web fonts alone. And since this piece is about how to read the symbols, writing them the same way the tables do is the more consistent choice.