Statistical data analyses
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1. Binomial, Poissonian and Gaussian distributions are particularly
important for statistical analyses in physics.
Could you explain what are relations between those, write down formulas
for probability distributions and point estimates: mean and standard
deviation.

2. When combining 1D measurements one often uses assumptions is that
they are distributed according to Gaussian distributions.
It leads to very commonly used expressions for calculating weighted
average and weighted error on the average.
Write down and explain expressions used. How it changes when included
are also correlations between measurements?
You can be qualitative in the answer.

3. You have measured  distributions of certain observable and have
theoretical model for that distribution. Explain how you would test
if the predictions are correct performing chi2 consistency test.
Write  down few formulas, do some illustrative drawings.
Would the chi2 method be correct in case of low statistics
distributions?

4. In some case more appropriate is to perform likelihood consistency
tests based on Poissonian statistics. Could you explain what does
it mean? Write  down few formulas, do some illustrative drawings.

5. Measurements have attributed statistical error and systematic
uncertainties.
Explain how one includes systematic uncertainties when performing
consistency tests with theoretical  model predictions or parameters
fitting.

6. Statistical analysis of physics measurement requires defining
first statistical model, which allows then to properly treat
statistical and systematic error.
Describe statistical model for a simple counting measurement,
with observable being number of events.

7. Statistical analysis of physics measurement requires defining
first statistical model, which allows then to properly treat
statistical and systematic error.
Describe statistical model for binned shape analysis,
with observable being n_i, i=1,..N_bins

8. Statistical analysis of physics measurement requires defining
first statistical model, which allows then to properly treat
statistical and systematic error.
Describe statistical model for unbinned shape analysis,
with observable being m_i, i=1,..N_events

9. Maximum Likelihood Estimation is nowadays a standard approach
to infer value of physical parameters (eg. couplings, masses)
from the measured distributions or event counts.
Could you explain this concept in case of binned shape analysis.

10. To claim discovery or exclusion based on specific measurement,
one first defines its statistical model, formulates hypotheses
H0 and H1 and then use likelihood ratio to decide between hypotheses.
Could you explain above procedures and compare/contrast
between discovery and exclusion (setting limits) cases.

11. To estimate expected results (limits) for a given measurement, one
is often generating special MC datasets (pseudo-experiments, toys).
Sometimes however is enough to generated so called Assimov data.
Could you explain the difference and how those datasets are used. 

12. Likelihood typically includes parameters of interest (POIs) and
nuisance parameters (NPs). Give example illustrating what they can be.
What about systematic uncertainties? How they can be incorporated
into likelihood. What does it mean "profiling systematic uncertainties"?