Constructing PDF vs VPDL

Useful References

1. Review of Basic Probability
2. Transforming Density Functions
3. CERN Data Analysis Briefbook

PDF transforms

1. construct two other variables: y2(x1,x2,x3), y3(x1,x2,x3)
2. find inverse functions: xi(y1,y2,y3)
3. PDF_y1,y2,y3 = |J| PDF_x1,x2,x3 (x1(y1,y2,y3),x2(y1,y2,y3),x3(y1,y2,y3))
• |J| = determinant of Jacobian Matrix = |d(x1,x2,x3)/d(y1,y2,y3)|

For our purposes:

• x1 = Lxy ; y1 = VPDL
• x2 = Pt(m) ; y2 = Pt(m)
• x3 = sigma(L) ; y3 = sigma(L)
• PDF(x1,x2,x3) = histogram

Histogram limits for transformation

Summary:
The problem here is really that the Physics Functions used in generating the PDF use VPDL and sigma(VPDL) instead of Lxy, sigma(Lxy) and Pt(m). This will be fixed in v5.0

Test this using c-cbar only sample (all fit variables fixed) = Set 6 in generation.

Generation

Variable	Source	N(bins)	Low	High
L(xy)	from func
Pt(m)	Bs_Dsmu.root	100	0	50
σ(L)	DLerr_Bs_JpsiPhi.root	50	0	0.02

Note: TH1::GetRandom() returns random number w/in bin - not just the bin center - so the binning used in generation is largely irrelevant.

Results for different PDF(x1,x2,x3) binning
Check RMS of VPDL distribution of VPDL(meas) in c-cbar events.
Default binning (set in life_fit.input):

Variable	N(bins)	Low	High
VPDL	150	-0.1	0.4
Lxy	150	-0.1	0.4
Pt(m)	150	0	30
σ(L)	30	0	0.015

Variable	N(bins)	Low	High	RMS	Plot
Data				0.01124
Default				0.00519	here
as Gen				0.00447	here
VPDL as Gen	50	-0.1	0.4	0.02514 but mean shifted	here
v5.0 VPDL L(xy) Pt(m) σ(L)	100 100 100 1	-0.1 -0.1 0 0.0035	0.4 0.4 50 0.0045	0.01071 use ave(σ)	here
VPDL L(xy) Pt(m) σ(L)	100 100 100 50	-0.1 -0.1 0 0	0.4 0.4 50 0.060	0.01022	here
σ(L)	50	0.001	0.020	0.00508
σ(L)	30	0.005	0.015	0.00551
σ(L)	30	0.008	0.015	0.00606
σ(L)	30	0.008	0.015	0.00648
σ(L)	30	0.015	0.025	0.00976
σ(L)	30	0.000	0.025	0.00764
σ(L)	30	0.000	0.030	0.00882	here