Normalization for PDFs that are a function of multiple variables

Formalism

Two Variables:
Joint pdf, f(x,y) is given in terms of the conditional pdf, f(x|y) (pdf of x given a fixed y).

• f(x,y) = f(x|y) f(y)

Assumption for Three Variables:
Considering y,z to be independent variables.

• f(x,y,z) = f(x|y,z) f(y|z) f(z) = f(x|y,z) f(y) f(z)

Lifetime/Mixing Fits Case

1. x = L_xy = L
2. y = P_t(meas) = P
3. z = σ(L) = σ

The PDFs for the independent variables are:

• f(P) = histogram of P_t(meas) for given source
• f(σ) = histogram of σ(L) - same for all sources
  • Note: this factorizes out of total PDF = sum of PDFs for all sources

The "physics" PDFs are:

• c-cbar Background:
  f(L|P,σ) = Gauss(L;σ) = (1/sqrt(2π)σ) exp(-L²/2σ²)
• Long Lifetime/Mixing
  f(L|P,σ) = ∫ dL^t R(L^t-L;σ) f_phys(L|P)
• f_phys
  
  • Lifetime: (M K / cτ P) exp(-(M K / cτ P) L^t)
  • Mixing: (1/2) (M K / cτ P) exp(-(M K / cτ P) L^t) (1 ± D cos( Δm t)
    where t = K M L / P