= Moments = '''Moments''' are measures of a distribution's shape and density. <> ---- == Description == The first raw moment is the mean: ''μ = E[X]''. For discrete variables, this is calculated as ''Σ x P(x=X)''; for continuous variables, as ''∫ x f(x) dx'' The second central moment is the variance: ''σ^2^ = E[(X - E[X])^2^] = E[(X - μ)^2^] = E(X^2^) - (E[X])^2^'' The derivation of this for discrete variables is: * ''Σ (x - μ)^2^ P(x=X)'' * ''Σ (x^2^ - 2μx + μ^2^) P(x=X)'' * ''Σ [x^2^ P(x=X)] - 2μ Σ [x P(x=X)] + μ^2^ Σ [P(x=X)]'' * ''[E[X^2^]] - 2μ [μ] + μ^2^ [1]'' * ''E[X^2^] - 2μ^2^ + μ^2^'' * ''E[X^2^] - μ^2^'' * ''E[X^2^] - (E[X])^2^'' The derivation of this for continuous variables is: * ''∫ (x - μ)^2^ f(x) dx'' * ''∫ (x^2^ - 2μx + μ^2^) f(x) dx'' * ''∫ [x^2^ f(x) dx] - 2μ ∫ [x f(x) dx] + μ^2^ ∫ [f(x) dx]'' * ''[E[X^2^]] - 2μ [μ] + μ^2^ [1]'' * ''E[X^2^] - 2μ^2^ + μ^2^'' * ''E[X^2^] - μ^2^'' * ''E[X^2^] - (E[X])^2^'' Through these derivations, it can be easily proven that (1) constants added to a variable do not affect variance, and (2) constant multipliers applied to a variable scale variance by their square. This is succinctly summarized as ''Var(aX + b) = a^2^ Var(X)'' The third central moment, skewness, measures lopsidedness of a distribution. The fourth central moment, kurtosis, measures the heaviness of the tails on a distribution. ---- == Errors == Models generally assume that individual errors average to zero, i.e. the first moment of errors is zero: ''E[Ŷ - Y] = 0''. Nonetheless, higher order moments are important. The '''mean square error''' ('''MSE''') is the second moment of the error: ''MSE(ˆθ) = E[(ˆθ - E[ˆθ])^2^]''. MSE can be decomposed into the variance of the estimator and bias: ''MSE(ˆθ) = Var(ˆθ) + Bias(ˆθ,θ)^2^ = Var(ˆθ) + (E[ˆθ]-θ)^2^''. Two important notes: * '''Bias''', i.e. ''E[ˆθ] - θ'', is ''not'' the same as the first moment of errors. * If there is no bias, then MSE ''is'' the variance of the estimator: ''MSE(ˆθ) = Var(ˆθ)''. ---- CategoryRicottone