ATLAS hints

The derivation of the terms in the correlation matrix

The covariance matrix is
[math]C_{ij} = \lt (x_i-\bar{x_i})(x_j-\bar{x_j})\gt [/math].
Suppose [math]\delta_1,\ \delta_2,\ \delta_3[/math] are three independent sources of normally-distributed unit fluctuations (with < δ_i > = 0 and < δ_i * δ_j > = δ_ij and where [math]\delta_{ij}[/math] is the kroneker delta function (1 for [math]i=j[/math] and zero for [math]i\neq j[/math]).

Pseudo-measurements of (x₁,x₂) can be generated from the expressions (x₁,x₂) = (x₁₀ + α * δ₁ + β * δ₂,x₂₀ + γ * δ₃ + λ * δ₂).

Expanding the covariance matrix one finds [math]C_{ij}=\lt x_i*x_j\gt -\bar{x_i}\bar{x_j}[/math].

Using the properties of [math]\lt \delta_i*\delta_j\gt [/math] above and [math]\lt x_i\gt =\bar{x_i}[/math], one finds [math]\lt x_1*x_2\gt = x_{10}*x_{20}+\beta*\lambda[/math]; so that C₁₂ = C₂₁ = β * λ. Similar substitution gives for the diagonal matrix elements
[math]C_{11}=x_{10}^2+\alpha^2+\beta^2-x_{10}^2[/math]
and
[math]C_{22}=x_{20}^2+\gamma^2+\lambda^2-x_{20}^2[/math]
so that C=

α2 + β2	β * λ
[math]\beta*\lambda[/math]	[math]\gamma^2+\lambda^2[/math]