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Greek Letters Common Usages Αα Alpha^ α: Constant in regression/statistics: y =^ α^ +^ βx +^ ε; also type I error Ββ Beta β: Coefficient in regression/statistics, often subscripted to indicate different coefficients: y = α + β 1 x 1 + β 2 x 2 + ε; also type II error; related, 1 – β is called the “power” of a statistical test. Γγ Gamma Γ: A particular statistical distribution; also used to denote a game. ∆δ Delta^ ∆: Means “change” or “difference”, as in the equation of a line’s slope: 1 2
1 2 x x
y y x
y −
.
δ: Known in game theory as the “discount parameter” and is used for repeated games. Εε Epsilon ε: “Error term” in regression/statistics; more generally used to denote an arbitrarily small, positive number. ∈ (Variant Epsilon)^ This version of epsilon is used in set theory to mean “belongs to” or “is in the set of”:^ x^ ∈^ X ; similarly used to indicate the range of a parameter: x ∈ [0, 1]. “ x ∉ ∅” means “the element x does not belong to the empty set”. Ζζ Zeta Ηη Eta Θθ Theta^ θ: The fixed probability of success parameter in a Binomial Distribution and related distributions. ϑ (Script Theta) Ιι Iota Κκ Kappa Λλ Lambda^ λ=^ n θ: Parameter in the Poisson Distribution. Μμ Mu^ μ: In statistics, the mean of a distribution. In game theory, often used as the probability of belief. Νν Nu Ξξ Xi Οο Omicron Ππ Pi^ ∏: Product symbol, as in 60
5
3
∏^ =
i =
i.
π: Mathematical constant (3.14159…); also used in game theory to denote an actor’s belief as a probability. Ρρ Rho^ ρ: Correlation coefficient in some statistical analyses. Σσ Sigma^ Σ: Summation symbol, as in 12
5
3
∑^ =
i =
i (^).
σ: Standard Deviation of a distribution; also used to denote an actor’s mixed strategy. σ^2 : Variance of a distribution. ς (Final Sigma) Ττ Tau Υυ Upsilon Φφ Phi^ Φ( z ): The cumulative density function (cdf) for the standard normal distribution. φ( z ): The probability density function (pdf) for the same. ϕ (Script Phi) Χχ Chi^ χ^2 : A particular statistical distribution. Ψψ Psi Ωω Omega Ω : The “positive definite matrix” in regression/statistics.
ϖ (Variant Omega)
Mathematical Constants e ≈ 2.718281828... π ≈ 3.141592653... i^ =^ −^1 (imaginary numbers)
Mathematical Symbols Usage
! Factorial^ ∏
=
=
n i
n i 1
! (^) ; e.g., 5! = 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 = 120
^ “carrot” or “hat” 3^2 = 3^2 = 9. Also used in statistics to denote estimates:^ σˆ
___ X x^ “bar”^ Sample mean of^ X^ = (Σ x )/ n , where^ n^ is the number of observations. ∀ All ∀ x ; for all^ x , something is true. ∃ Exists ∃ x = 1; there exists some x equal to 1. → Implies p → q ; if p is true (or occurs), then q is true (or will occur). ∴ Therefore^ Indicating a logical result: p → q and q → r , ∴ p → r. | Given, Conditional^ P( E | F ); The probability of^ E^ given (or within the set of)^ F. | | Absolute Value |– x | = x ∼ Not ~C; not to cooperate. (Also used in geometry to mean “similar”.) ≤ Less than or equal to ≥ Greater than or equal to ∞ Infinity ± Plus or minus ∝ Proportional to x ∝ 1/ f ∂ Derivative Calculus notation;^ ∂ x (^ y = mx + b )^ = m
∂
≠ Not equal to ≡ Identically equal to^ x^ ≡^ x ; sometimes a way proving something; also a way of denoting a definition. ≈ Approximately equal to^ π^ ≈^ 3. ℜ, R , or (^) Set of Real Numbers
∅ Empty Set^ X^ =^ ∅; The set^ X^ is empty.^ X^ ≠^ ∅; The set^ X^ is not empty. ∩ Conjunction; And^ {1, 2, 3, 4}^ ∩^ {4, 5, 6, 7} = {4} ∪ Union; Or {1, 2, 3, 4}^ ∪^ {4, 5, 6, 7} = {1, 2, 3, 4, 5, 6, 7} √ Square root 2 ≈ 1.414; 4 = 2 P(⋅) Probability of P(HH) = ½ ⋅ ½ = ¼; the probability of landing two heads in successive coin flips; sometimes Pr(⋅). L Likelihood Used in Maximum Likelihood Estimation in statistics. L(⋅) Lottery^ (in game theory)^ L( B , W ; p ) is a lottery between winning one’s best outcome, B , with probability p and “winning” one’s worst outcome, W , with probability 1 – p. E(⋅) Expectation of E( X ) =^ Σ^ x ⋅^ P( x ); also as expected utility: EU(L(1, 0; ¼)) = 1^ ⋅^ ¼ + 0^ ⋅^ ¾ = ¼. ln or LN Natural log (ln( x ) = b )^ ≡^ (eb^ = x), where e is the mathematical constant.
lim Limit 0 lim 1 = x →∞ (^) x ; The limit of 1/ x^ as^ x^ goes to (or approaches) infinity equals zero.
∫ Integral^ Calculus notation;^ ∫^ =^
2
x dx 21 x and ∫ +
= −
b a a b
dx b a a b
1
J Jacobian J : a particular matrix; J : Determinant of a Jacobian matrix.
