Addition and convolution: extended maximum likelihood fit with alternate range definition for observed number of events.

**Author:** Clemens Lange, Wouter Verkerke (C++ version)

*This notebook tutorial was automatically generated with ROOTBOOK-izer from the macro found in the ROOT repository on Monday, January 17, 2022 at 09:53 AM.*

In [ ]:

```
import ROOT
```

Declare observable x

In [ ]:

```
x = ROOT.RooRealVar("x", "x", 0, 10)
```

Create two Gaussian PDFs g1(x,mean1,sigma) anf g2(x,mean2,sigma) and their parameters

In [ ]:

```
mean = ROOT.RooRealVar("mean", "mean of gaussians", 5)
sigma1 = ROOT.RooRealVar("sigma1", "width of gaussians", 0.5)
sigma2 = ROOT.RooRealVar("sigma2", "width of gaussians", 1)
sig1 = ROOT.RooGaussian("sig1", "Signal component 1", x, mean, sigma1)
sig2 = ROOT.RooGaussian("sig2", "Signal component 2", x, mean, sigma2)
```

Build Chebychev polynomial pdf

In [ ]:

```
a0 = ROOT.RooRealVar("a0", "a0", 0.5, 0.0, 1.0)
a1 = ROOT.RooRealVar("a1", "a1", -0.2, 0.0, 1.0)
bkg = ROOT.RooChebychev("bkg", "Background", x, [a0, a1])
```

Sum the signal components into a composite signal pdf

In [ ]:

```
sig1frac = ROOT.RooRealVar("sig1frac", "fraction of component 1 in signal", 0.8, 0.0, 1.0)
sig = ROOT.RooAddPdf("sig", "Signal", [sig1, sig2], [sig1frac])
```

Define signal range in which events counts are to be defined

In [ ]:

```
x.setRange("signalRange", 4, 6)
```

Associated nsig/nbkg as expected number of events with sig/bkg
_in_the*range* "signalRange"

In [ ]:

```
nsig = ROOT.RooRealVar("nsig", "number of signal events in signalRange", 500, 0.0, 10000)
nbkg = ROOT.RooRealVar("nbkg", "number of background events in signalRange", 500, 0, 10000)
esig = ROOT.RooExtendPdf("esig", "extended signal pdf", sig, nsig, "signalRange")
ebkg = ROOT.RooExtendPdf("ebkg", "extended background pdf", bkg, nbkg, "signalRange")
```

Construct sum of two extended pdf (no coefficients required)

In [ ]:

```
model = ROOT.RooAddPdf("model", "(g1+g2)+a", [ebkg, esig])
```

Generate 1000 events from model so that nsig, come out to numbers <<500 in fit

In [ ]:

```
data = model.generate({x}, 1000)
```

Perform unbinned extended ML fit to data

In [ ]:

```
r = model.fitTo(data, Extended=True, Save=True)
r.Print()
```