# Presentation: 305873

## Statistics:

**Basic Concepts**

- Aneta Siemiginowska
- Harvard-Smithsonian Center for Astrophysics

## OUTLINE

- Motivation: why do we need statistics?
- Probabilities/Distributions
- Poisson Likelihood
- Parameter Estimation
- Statistical Issues

## Why do we need Statistics?

- How do we take decisions in Science?
- _Tools:_ instruments, data collections, reduction, classifications – tools and techniques
- _Decisions:_ is this hypothesis correct? Why not? Are theses data consistent with other data? Do we get an answer to our question? Do we need more data?
- _Comparison to decide:_
- Describe properties of an object or sample:

- _Example:_
-  Is a faint extension
- a jet or a point source?

**GB 1508+5714 z=4.3**

*Siemiginowska et al (2003)**‏*

## Stages in Astronomy Experiments

**OBSERVE****	****Carefully****	****Experiment design,****	****What? Number **
**	****	****	****	****exposure time****	**** (S)****	****of objects, Type? ****(S)****‏**
**REDUCE****	****Algorithms****	****calibration files****	****data quality**
**	****	****	****	****QE,RMF,ARF,PSF ****(S)****	****Signal-to-Noise**** (S)**** ****	****	****	****	****	****	****	**** **** **
**ANALYSE ****	****Parameter****	****Intensity, positions****	****Frequentist**
**	****	****Estimation,****	**** **** (S)****	****	****	****Bayesian?**
**	****	****Hypothesis**
**	****	****testing****	****(S)                                              (S)****‏**
**CONCLUDE       Hypothesis****	****Distribution tests,****	****Belivable,**
**	****	****testing****	**** (S) ****	****Correlations ****(S)****	****Repeatable,**
**	****	****	****	****	****	****	****Understandable? ****(S)****‏**
**REFLECT****	****Carefully****	****Mission achieved?****	****The next **
**	****	****	****	****A better way?****	****	****Observations ****(S)****‏**
**	****	****	****	****We need more data!**
**	****	****	****	****(S)****‏**

-  **Stage****	****	****How****	****	****Example****	****	****Considerations**

- Wall & Jenkins (2003)‏

##  Statistic is a quantity that summarizes data

**=> Astronomers cannot avoid Statistics**

- Statistics are combinations of data that do not depend on unknown parameters:
- Mean, averages from multiple experiments etc.

## Probability

- Numerical formalization of our degree of belief.

- Number of favorable events
-    Total number of  events

- _Example 2: _
- Use data to calculate probability, thus the probability of a cloudy observing run:
- number of cloudy nights last year
- 	365 days
- _Issues:_
-  limited data
-  not all nights are *equally likely* to be cloudy

**Laplace principle of indifference:**
**  **
**All events have equal probability**
** **
- _Example 1:_
- 1/6 is the probability of throwing a 6 with 1 roll of the dice **BUT** the dice can be biased! =>  need to calculate the probability of each face

****

## Conditionality and Independence

-  A and B events are **independent** if the probability of one is unaffected by what we know about the other:
- 	prob(A and B)=prob(A)prob(B)‏

- If the probability of A depends on what we know about B 	   A given B    =>  **conditional probability**
**	****                           **prob(A and B)‏
**                  **prob(A|B)**=**
**	****	****	****   ** prob(B)‏

- If A and B are independent => prob(A|B)=prob(A)‏
- If there are several possibilities for event B (B1, B2....)‏
- 	   prob(A) = ∑prob(A|Bi) prob(Bi)‏
- A – parameter of interest
- Bi – not of interest, instrumental parameters, background
- prob(Bi) 	- if known we can sum (or integrate) -* Marginalize*

## Bayes' Theorem

- Bayes' Theorem is derived by equating:
- 		prob(A and B) = prob (B and A)‏
- 	             **    ****prob (A|B) prob(B)****‏**
-          **prob(B|A) =**
- 	                        **prob(A)****‏**
- _Gives the Rule for induction: _
-    the data, the event A,  are succeeding B, the state of belief preceeding
-    the experiment.
-  prob(B) – **prior probability** which will be modified by experience
-  prob(A|B) – **likelihood **
** **prob(B|A) **– posterior probability **– the state of belief after the 					data have been analyzed
-  prob(A) – normalization

## Example

- _A box with colored balls: _
-           what is the content of the box?
-    prob(content of the box | data) ∝ prob(data | content of the box)‏

- _Experiment:_
- 	 N red balls
-  	 M white balls
-  	 N+M = 10 total, known
-  Draw 5 times (putting back) (T) and
-  get 3 red balls (R)‏
-  *How many red balls are in the box?*

-  Model (our hypothesis) =>                       N
- 			prob(R) =
- 				      N+M

-   Likelihood = (   ) prob(R)R prob(M)T-R

- T=5
- T=50

## Example

**From a paper by Martinez-Sansigre et al published in **
**Aug 4, 2005 issue of** *Nature*

- q – quasar fraction
-        Type-1 quasars         N1
- q=                              =
-       Type-1 + Type-2    N1+N2

- <N1> - number of Type-1 qso
- <N2> - number of Type-2 qso
- 1/ take Poisson likelihood with the mean <N2> = (1-q)<N1>/q
- 2/ evaluate likelihood at each q and N1
- 3/ integrate P(N1|q)P(N1) over N1

- Posterior Probability distribution for the quasar fraction

- What is the fraction of the       unobscured quasars?
- Use new Spitzer observations

**p(q|data,{type-1 qso}) = p(data|q,{type-1 qso})****‏**

- Include only 5 qso

- 5+6 qso

- Torus Models

## Probability Distributions

**Probability  is crucial in decision process:**** **
_**Example: **_
** **
**Limited data yields only partial idea about the line width in the spectrum. We can only assign the probability to the range of the line width roughly matching this parameter. We decide on the presence of the line by calculating the probability.**

## Definitions

***Random variable:*** a variable which can take on different numerical values, corresponding to different experimental outcomes.
- Example: a binned datum *D**i* , which can have different values even when an experiment is repeated exactly.
***Statistic:*** a function of random variables.
- Example: a datum *D**i** *, or a population mean
***Probability sampling distribution:*** the normalized distribution from which a statistic is sampled. Such a distribution is commonly denoted *p** *(*X** *| *Y** *), “the probability of outcome X given condition(s) *Y*,” or sometimes just *p** *(*X** *). Note that in the special case of the Gaussian (or normal) distribution, *p** *(*X** *) may be written as *N*(*μ*,*** *2), where *μ** *is the Gaussian mean, and *** *2 is its variance.

## The Poisson Distribution

-       prob(Di)=
- probability of finding *D**i* events (*counts*) in bin *i* (*energy rage*) of dataset *D *(*spectrum*) in a given length of time (exposure time), if the events occur independently at a constant rate *M**i  **(source intensity).*

_**The Poisson Distribution**_

- Things to remember:
*Mean**   ****μ*****= *****E *****[*****D******i*****] = *****M******i***
- Variance:  ** V [*****D******i*****] = *****M******i***
**cov[*****D******i*****1**** ****, *****D******i*****2**] = 0 => independent
- the sum of *n* Poisson-distributed variables is itself Poisson-distributed with variance:

*Collecting X-ray data  => Counting individual photons *
*                 **=> Sampling from Poisson distribution*

- As *M**i* =>   Poisson distribution
- converges to Gaussian distribution
-  N(*μ* = *M**i* ; **2 = *M**i* )‏

## Example:

- Integer counts spectrum sampled from a constant amplitude model with mean *μ *= 60 counts, and fit with a parabolic model.

_**Example:**_

## Example2

- Example of a two-dimensional integer counts spectrum.  *Top Left: Chandra* ACIS-S data of X-ray cluster  MS 2137.3-2353, with *ds9* source regions superimposed.
*Top Right:* Best-fit of a two-dimensional beta model to the filtered data.
*Bottom Left:* Residuals (in units of *** *) of the best fit.
*Bottom Right:* The applied filter; the data within the ovals were excluded from the fit.

_**Example2**_

## Poisson vs. Gaussian Distributions – Low Number of Counts

- Comparison of Poisson distributions (dotted) of mean *μ* = 2 and 5 with normal distributions of the same mean and variance (Eadie *et al.* 1971, p. 50).

_**Poisson vs. Gaussian Distributions – Low Number of Counts**_

**μ=2**

**μ=5**

## μ=10

- Comparison of Poisson distributions (dotted) of mean *μ* = 10, 25 and 40 with normal distributions of the same mean and variance (Eadie *et al.* 1971, p. 50).

**μ=****10**

**μ=25**

**μ=40**

_**Poisson vs. Gaussian Distributions**_

## Gaussian Distribution

- For large *μ*->∞ Poisson (and the Binomial, large T) distributions converge to Gaussian (normal) distributions.

*	**         **1*
*prob(x) =             **exp[-(x-μ)**2**/2σ**2**]*
*	**      **σ√2π *

- Mean - *μ*
- Variance - *σ**2*

*Note:  Importance of the Tails!*
- +/-*2**σ* range covers *95.45%* of the area, so *2σ* result has less than *5%* chance of occurring by chance, but because of the error estimates this is not the acceptable result. Usually *3σ* or *10σ*  have to be quoted and the convergence to Gaussian fastest in the center than in the tails!

## Central Limit Theorem

**The true importance of the Gaussian distribution**

- Form averages *M**n * from repeated drawing of *n* samples from a population with finite mean *μ* and variance *σ**2*
*	**(M**n**-μ)**‏*
*          *
*	** **σ/√n**	**         as n→∞            *
* **	**	**      **μ=0, σ**2**=1*

*=> Gaussian Distribution*

- single

- averages of 4

- averages of 2

- averages of  16

- 200 y values drawn from exp(-x) function

## Bayesian vs. Classical 

_**Classical Approach: **_
- Assuming the true distance *D**0*  then *D* is normally distributed around  *D**0*  with a standard deviation of 0.1. Repeating measurement will yield many estimates of distance* D*  which all scatter around true *D**0.*
*Assume the thing (distance) we want to know and tell us *
*how the data will behave.*

_**Bayesian Approach:**_
- Deduce directly the probability distribution of *D**0* from the data. *Assumes the data and tell us the thing we want to know. No repetition of experiment.*

_**Example:**__ _
-               **D = 8.5****∓****0.1 Mpc **
- Does not describe probability that a true value is between 8.4 and 8.6.
- We **assume** that a Gaussian distribution applies and knowing the distribution of errors we can make probabilistic statements.

## What do we really do?

_**Example:**_
- I've observed my source, reduce the data and finally got my X-ray spectrum –  what do I do now? How can I find out what does the spectrum tell me about the physics of my source?
- Run **XSPEC** or *Sherpa*! But what do those programs really do?

- Chandra ACIS-S

_**Fit the data**__ _=> **C(h)****=****∫****R(E,h) A(E) M(E,****θ****)dE**** **
- Assume a model and look for the best model parameters which  describes the observed spectrum.

**Need a Parameter Estimator - Statistics**

- Counts

- Response
-      Effective Area

- Model

- h- detector channels
- E- Energy
- θ- model parameters

## Parameter Estimators

- Large variance

- Best

- Biased

*θ**0*

- Statistic

_**Requirements on Statistics:**_
** ****Unbiased  **
**- converge to true value with repeated measurements**
** ****Robust **
**– ****less affected by outliers**
** ****Consistent **
**– ****true value for a large sample size (Example: rms and Gaussian distribution)****‏**
** ****Closeness**** **
**- smallest variations from the truth**

## One can use the Poisson distribution to assess the probability of sampling a datum Di given a predicted (convolved) model amplitude Mi. Thus to assess the quality of a fit, it is natural to maximize the product of Poisson probabilities in each data bin, i.e., to maximize the Poisson likelihood: 

In practice, what is often maximized is the log-likelihood, 
L = logL. A well-known statistic in X-ray astronomy which is related to L is the so-called “Cash statistic”: 

- In practice, what is often maximized is the log-likelihood,
*L* = logL. A well-known statistic in X-ray astronomy which is related to *L* is the so-called “Cash statistic”:

_**Maximum Likelihood:**__**Assessing the Quality of Fit**_

**∝**

## If the hypothesized  is close to the true value, then we expect 

- If the hypothesized ** is close to the true value, then we expect
- a high probability to get data like that which we actually found.

- So we define the maximum likelihood (ML) estimator(s) to be
- the parameter value(s) for which the likelihood is maximum.

## (Non-) Use of the Poisson Likelihood

- In model fits, the Poisson likelihood is not as commonly used as it should be. Some reasons why include:
-  	a historical aversion to computing factorials;
-  	the fact the likelihood cannot be used to fit “background subtracted” spectra;
-  	the fact that negative amplitudes are not allowed (not a bad thing physics abhors negative fluxes!);
-  	the fact that there is no “goodness of fit" criterion, i.e. there is no easy way to interpret Lmax (however, *cf*. the **CSTAT** statistic); and
-  	the fact that there is an alternative in the Gaussian limit: the 2 statistic.

## 2   Statistic

- Definition:   2= ∑i (*D**i**-M**i*)2/*M**i*

- The  2 statistics is **minimized** in the fitting the data, varying the model parameters until the best-fit model parameters are found for the minimum value of the
- 2 statistic

- Degrees-of-freedom = **k-1- N**
- N – number of parameters
- K – number of spectral bins

## Confidence Limits

- Essential issue = after the bets-fit parameters are found estimate the confidence limits for them. The region of confidence is given by (Avni 1976):
- 			 2**** = 2min +(,)
-    - degrees of freedom
-    - significance
-   2min  - minimum
* ** **** **depends only on the number of *
*         **parameters involved*
*        **nor on goodness of fit*

**Significance      Number of parameters**
**    ******** ****	****	****    ****1      2        3**
** **
** ****0.68****	****	**** 1.00  2.30   3.50**
** ****0.90****	****	**** 2.71  4.61   6.25**
** ****0.99 ****	****	**** 6.63  9.21   11.30   **

## CalculatingConfidence Limits means  Exploring  the Parameter Space - Statistical Surface

- Example of a “well-behaved” statistical surface in parameter space, viewed as a multi-dimensional paraboloid (**2, *top*), and as a multi-dimensional Gaussian (exp(-**2** **/2) ≈ L, *bottom*).

**CalculatingConfidence Limits means  Exploring  the Parameter Space - Statistical Surface**

## Behaviour of Statistics for One Parameter

**Behaviour of Statistics for One Parameter**

- Comparison of Two methods in Sherpa

## Confidence Limits for Two Parameters

**Confidence Limits for Two Parameters**

*1**σ**, 2**σ**, 3**σ contours*

*+ Best fit parameters*

- Comparison of Two methods in Sherpa

## The version of 2 derived above is dubbed “data variance” 2 , or     , because of the presence of D in the denominator. Generally, the 2 statistic is written as: 

where      represents the (unknown!) variance of the Poisson distribution from which Di is sampled. 
	  	2 Statistic		
		Data Variance		 Di 
		Model Variance		 Mi
		Gehrels		    [1+(Di+0.75)1/2]2
		Primini		 Mi from previous best-fit
		Churazov	 based on smoothed data D
		“Parent”
		Least Squares		1

Note that some X-ray data analysis routines may estimate i for you during data reduction. In PHA files, such estimates are recorded in the STAT_ERR column. 

- where      represents the (unknown!) variance of the Poisson distribution from which *D**i *is sampled.
- 	  	******2 ****Statistic**
- 		Data Variance		 *D**i *
*	**	*Model Variance		 *M**i*
- 		Gehrels		    [1+(Di+0.75)1/2]2
- 		Primini		 *M**i *from previous best-fit
- 		Churazov	 based on *smoothed* data D
- 		“Parent”
- 		Least Squares		1
- Note that some X-ray data analysis routines may estimate ***i* for you during data reduction. In PHA files, such estimates are recorded in the **STAT_ERR** column.

**“****Versions” of the**** ********2 ****Statistic**

## Statistical Issues

-  Bias
-  Goodness of Fit
-  Background Subtraction
-  Rebinning
-  Errors

## Statistical Issues: Bias 

- If one samples a large number of datasets from a given model        and then fits this same model to these datasets (while letting ***  *vary), one will build up sampling distributions for each parameter ***k* .
- An estimator (*e.g**., *2) is biased if the mean of these distributions (E[***k*]) differs from the true values ***k,o*.
- The Poisson likelihood is an unbiased estimator.
- The 2 statistic *can* be biased, depending upon the choice of *** *:
- Using the *Sherpa* utility **FAKEIT**, we simulated 500 datasets from a constant model with amplitude 100 counts.
- We then fit each dataset with a constant model, recording the inferred amplitude.
**	****	****	****Statistic****	****   ****	****Mean Amplitude**
**	****	****	**Gehrels	                           99.05
- 			Data Variance	           99.02
- 			Model Variance	         100.47
- 			“Parent”   	                           99.94
- 			Primini		           99.94
- 			Cash		           99.98

## A demonstration of bias.  Five hundred datasets are sampled from a constant model with amplitude 100 and then are fit with the same constant amplitude model, using 2 with data variance. The mean of the distribution of fit amplitude values is not 100, as it would be if the statistic were an unbiased estimator. 

- A demonstration of bias.  Five hundred datasets are sampled from a constant model with amplitude 100 and then are fit with the same constant amplitude model, using 2 with data variance. The mean of the distribution of fit amplitude values is not 100, as it would be if the statistic were an unbiased estimator.

##  The 2 goodness-of-fit is derived by computing

  This can be computed numerically using, e.g., the GAMMQ routine of Numerical Recipes. 
A typical criterion for rejecting a model is    < 0.05   (the “95% criterion”). However, using this criterion blindly is not recommended! 
A quick’n’dirty approach to building intuition about how well your model fits the data is to use the reduced 2, i.e., 

A “good” fit has 2 obs,r 1
If        0 the fit is “too good” -- which means (1) the errorbars are too large, (2)   2 obs,r  is not sampled from the 2 distribution, and/or (3) the data have been fudged.
  The reduced 2 should never be used in any mathematical computation if you are using it, you are probably doing something wrong! 

-   This can be computed numerically using, *e.g.*, the **GAMMQ** routine of *Numerical Recipes*.
- A typical criterion for rejecting a model is    < 0.05   (the “95% criterion”). However, using this criterion blindly *is not recommended*!
- A quick’n’dirty approach to building intuition about how well your model fits the data is to use the *reduced* 2, *i.e.,*
- A “good” fit has 2 obs,r 1
- If        0 the fit is “too good” -- which means (1) the errorbars are too large, (2)   2 obs,r  is *not* sampled from the 2 distribution, and/or (3) the data have been fudged.
-   The reduced 2 should never be used in any mathematical computation if you are using it, you are probably doing something wrong!

_**Statistical Issues: Goodness-of-Fit**_

## Figure 7: Comparison of the distributions of 500 sampled values of 2 versus the expected distribution for 99 degrees of freedom. Top: 2 with Gehrels variance. Bottom: 2 with data variance. 

- Figure 7: Comparison of the distributions of 500 sampled values of 2** **versus the expected distribution for 99 degrees of freedom. Top: 2** **with Gehrels variance. Bottom: 2** **with data variance.** **

## Slide 37

## Figure 8: Top: Best-fit of a power-law times galactic absorption model to the source spectrum of supernova remnant G21.5-0.9. Bottom: Best-fit of a separate power-law times galactic absorption model to the background spectrum extracted for the same source. 

- Figure 8: *Top:* Best-fit of a power-law times galactic absorption model to the source spectrum of supernova remnant G21.5-0.9. *Bottom:* Best-fit of a separate power-law times galactic absorption model to the background spectrum extracted for the same source.

## Statistical Issues: Background Subtraction

- Why subtract the background?
- It may be difficult to select an appropriate model shape for the background.
- Analysis proceeds faster, since background datasets are not fit.
- “It won't make any difference to the final results.”
- Why not subtract the background?
- The data    are not Poisson-distributed -- one cannot fit them with the Poisson likelihood. (Variances are estimated via *error propagation*:
- It may well make a difference to the final results:
- Subtraction reduces the amount of statistical information in the analysis quantitative accuracy is thus reduced.
- Fluctuations can have an adverse effect, in, *e.g.*, *line detection*.

## Statistical Issues: Rebinning 

*Rebinning data invariably leads to a loss of statistical information!*
- Rebinning is not necessary if one uses the Poisson likelihood to make statistical inferences.
- However, the rebinning of data may be necessary to use 2 statistics, if the number of counts in any bin is <= 5. In X-ray astronomy, rebinning (or *grouping*) of data may be accomplished with:
**grppha**, an *FTOOLS* routine; or
**dmgroup**, a *CIAO* Data Model Library routine.

- One common criterion is to sum the data in adjacent bins until the sum equals five (or more).

- Caveat: always estimate the errors in rebinned spectra using the new data     in each new bin (since these data are still Poisson-distributed), rather than propagating the errors in each old bin.
-  For example, if three bins with numbers of counts 1, 3, and 1 are grouped to make one bin with 5 counts, one should estimate *V*[*D*’= 5] and *not* *V*[*D*’] = *V*[*D*1 = 1] + V[D2 = 3] + V [D3 = 1]. The propagated errors may overestimate the true errors.

## Statistical Issues: Systematic Errors 

- In X-ray astronomy, one usually speaks of two types of errors: statistical errors, and systematic errors.
- Systematic errors are uncertainties in instrumental calibration. For instance:
- Assume a  spectrum observed for time *t* with a telescope with perfect resolution and an effective area *A**i* . Furthermore, assume that the uncertainty in *A**i* is ***A,i* .
- Neglecting data sampling, in bin *i*, the expected number of counts is     *D**i** *= *D****,i*(*E *)*tA**i**.*
- We estimate the uncertainty in *D**i** *as
*	** ****D**i* = *D****,i*(*E *)*t****A,I* = *D****,i*(*E *)*tf**i**A**i *= *f**i**D**i*
- The systematic error *f**i**D**i *; in PHA files, the quantity *f**i *is recorded in the SYS_ERR column.
- Systematic errors are added in quadrature with statistical errors; for instance, if one uses        to assess the quality of fit, then  i = *(D**i**+D**i**f**i**)**1/2*
- To use information about systematic errors in a Poisson likelihood fit, one must incorporate this information into the model, as opposed to simply adjusting the estimated error for each datum.

## Drake et al 2006, SPIE meeting

## Slide 43

## Summary

- Motivation: why do we need statistics?
- Probabilities/Distributions
- Poisson Likelihood
- Parameter Estimation
- Statistical Issues
- Statistical Tests – still to come....

## Conclusions

**Statistics is the main tool for any astronomer who need to do data analysis and need to decide about the physics presented in the observations.**

- _References:_
- Peter Freeman's Lectures from the Past X-ray Astronomy School
- _[ ](http://xrayschool.gsfc.nasa.gov/docs/xrayschool-2003/talks.html)_
- “Practical Statistics for Astronomers”, Wall & Jenkins, 2003
-  Cambridge University Press
- Eadie et al 1976, “Statistical Methods in Experimental Physics”

## General statistics:

- Babu, G. J., Feigelson, E. D. 1996, Astrostatistics (London: Chapman & Hall)‏
- Eadie, W. T., Drijard, D., James, F. E.,Roos, M., & Sadoulet, B. 1971, Statistical Methods in Experimental Physics (Amsterdam: North-Holland)‏
- Press, W. H., Teukolsky, S. A., Vetterling, W. T.,& Flannery, B. P. 1992, Numerical Recipes (Cambridge: Cambridge Univ. Press)‏
- Introduction to Bayesian Statistics:
- Loredo, T. J. 1992, in Statistical Challenges in Modern Astronomy,ed. E. Feigelson & G. Babu (New York: Springer-Verlag), 275
- Modified Land 2 *s*tatistics:
- Cash, W. 1979, ApJ 228, 939\item Churazov, E., et al. 1996, ApJ 471, 673
- Gehrels, N. 1986, ApJ 303, 336
-  Kearns, K., Primini, F., & Alexander, D. 1995, in Astronomical Data Analysis Software and Systems IV,eds. R. A. Shaw, H. E. Payne, & J. J. E. Hayes (San Francisco: ASP), 331
- Issues in Fitting:
- Freeman, P. E., et al. 1999, ApJ 524, 753 (and references therein)‏
*Sherpa* and *XSPEC*:
- Freeman, P. E., Doe, S., & Siemiginowska, A. 2001, astro-ph/0108426
- http://asc.harvard.edu/ciao/download/doc/sherpa_html_manual/index.html
- Arnaud, K. A. 1996, in Astronomical Data Analysis Software and Systems V, eds. G. H. Jacoby & J. Barnes (San Francisco: ASP), 17
- http://heasarc.gsfc.nasa.gov/docs/xanadu/xspec/manual/manual.html

## Properties of Distributions

***The beginning X-ray astronomer only needs to be familiar with four properties of distributions: the mean, mode, variance, and standard deviation, or “error.”***
*** ******Mean: ******  μ = E[X ] =******  ******∫******dX X p(X)******‏***
-  Mode:  max[p(X)]
-  Variance:
-  Error:
- Note that if the distribution is Gaussian, then σ is indeed the Gaussian * **σ** *(hence the notation).
- If two random variables are to be jointly considered, then the sampling distribution is two-dimensional, with shape locally described by the *covariance matrix*:
-                                                              where
- The related *correlation coefficient* is
- The correlation coefficient can range from -1 to 1.

## Properties of Probability

- Formalize the “measure of belief”:
- A,B,C – three events and we need to measure how strongly we think each is likely to happen and apply the rule:
**If A is more likely than B, and B is more likely than C, then A is more likely than C**.

-  Any random event A has a probability
-          **prob(A) between 0 and 1**
- The sure event **prob(A) = 1**
- If A and B are exclusive (**A****∩****B=0),  **disjoint** **events then              **prob(A**** or ****B) = prob(A) +prob(B)****‏**

_**Kolmogorov axioms – **_
**              ****Fundation of the Theory of Probability**

## Statistics

**(1) Statistics indicating the location of the data: **
** **
-    Average:   <X> = (1/N) ∑i  Xi
-    Mode: location of the peak in the histogram; the value 			occuring most frequently
**(2) Statistics indicating the scale or amount of scatter:**
**   **Mean deviation: <ΔX> = (1/N) ∑i  |Xi -<X>|
-    Mean square deviation: S2 = (1/N) ∑i  (Xi - <X>)2
-      Root Mean Square deviation: rms = S