282 Monte Carlo
For simplicity, we introduce the notation
∫
dxφ(x)f(x) =〈φ〉f, (7.2.5)where〈···〉findicates an average ofφ(x) with respect to the distributionf(x). We
wish to compute the probabilityP(y) that the estimatorI ̃Mwill have a valuey. This
probability is given formally by
P(y) =∫
dx 1 ···dxM[M
∏
i=1f(xi)]
δ(
1
M
∑M
i=1φ(xi)−y)
, (7.2.6)
where the Diracδ-function restricts the integral to those sets of vectors x 1 ,...,xMfor
which the estimator is equal toy. Eqn. (7.2.6) can be simplified by introducing the
integral representation of theδ-function (see Appendix A)
δ(z) =1
2 π∫∞
−∞dσeizσ. (7.2.7)Substituting eqn. (7.2.7) into eqn. (7.2.6) and using the general property ofδ-functions
thatδ(ax) = (1/|a|)δ(x) yields
P(y) =M∫
dx 1 ···dxM[M
∏
i=1f(xi)]
δ(M
∑
i=1φ(xi)−My)
=
M
2 π∫
dx 1 ···dxM[M
∏
i=1f(xi)]∫
∞−∞dσe
iσ(∑M
i=1φ(xi)−My)
. (7.2.8)
Interchanging the order of integrations gives
P(y) =M
2 π∫∞
−∞dσe−iMσy∫
dx 1 ···dxM[M
∏
i=1f(xi)]
eiσ∑M
i=1φ(xi)=
M
2 π∫∞
−∞dσe−iMσy[∫
dxf(x)eiσφ(x)]M
=
M
2 π∫∞
−∞dσe−iMσyeMln∫
dxf(x)eiσφ(x)=
M
2 π∫∞
−∞dσeMF(σ,y), (7.2.9)where in the second line, we have used the fact that the integrals over x 1 , x 2 ,... in the
product are all identical. In the last line of eqn. (7.2.9), the functionF(σ,y) is defined
to be
F(σ,y) =−iσy+g(σ) (7.2.10)