# Statistics/Probability/Combinatorics

Statistics
* Introduction
** What Is Statistics?
** Subjects in Modern Statistics
** Why Should I Learn Statistics?
** What Do I Need to Know to Learn Statistics?
* Different Types of Data
** Primary and Secondary Data
** Quantitative and Qualitative Data
* Methods of Data Collection
** Experiments
** Sample Surveys
** Observational Studies
* Data Analysis
** Data Cleaning
** Moving Average
* Summary Statistics
** Measures of center
*** Mean, Median, and Mode
*** Geometric Mean
*** Harmonic Mean
*** Relationships among Arithmetic, Geometric, and Harmonic Mean
*** Geometric Median
** Measures of dispersion
*** Range of the Data
*** Variance and Standard Deviation
*** Quartiles and Quartile Range
*** Quantiles
* Displaying Data
** Bar Charts
** Comparative Bar Charts
** Histograms
** Scatter Plots
** Box Plots
** Pie Charts
** Comparative Pie Charts
** Pictograms
** Line Graphs
** Frequency Polygon
* Probability
** Combinatorics
** Bernoulli Trials
** Introductory Bayesian Analysis
* Distributions
** Discrete Distributions
*** Uniform Distribution
*** Bernoulli Distribution
*** Binomial Distribution
*** Poisson Distribution
*** Geometric Distribution
*** Negative Binomial Distribution
*** Hypergeometric Distribution
** Continuous Distributions
*** Uniform Distribution
*** Exponential Distribution
*** Gamma Distribution
*** Normal Distribution
*** Chi-Square Distribution
*** Student-t Distribution
*** F Distribution
*** Beta Distribution
*** Weibull Distribution
* Testing Statistical Hypothesis
** Purpose of Statistical Tests
** Formalism Used
** Different Types of Tests
** z Test for a Single Mean
** z Test for Two Means
** t Test for a single mean
** t Test for Two Means
** paired t Test for comparing Means
** One-Way ANOVA F Test
** z Test for a Single Proportion
** z Test for Two Proportions
** Testing whether Proportion A Is Greater than Proportion B in Microsoft Excel
** Spearman's Rank Coefficient
** Pearson's Product Moment Correlation Coefficient
** Chi-Squared Tests
*** Chi-Squared Test for Multiple Proportions
*** Chi-Squared Test for Contingency
** Approximations of distributions
* Point Estimates (12:07, 28 March 2007 (UTC))
** Unbiasedness
** Measures of goodness
** UMVUE
** Completeness
** Sufficiency and Minimal Sufficiency
** Ancillarity
* Practice Problems
** Summary Statistics Problems
** Data-Display Problems
** Distributions Problems
** Data-Testing Problems
* Numerical Methods
** Basic Linear Algebra and Gram-Schmidt Orthogonalization
** Unconstrained Optimization
** Quantile Regression
** Numerical Comparison of Statistical Software
** Numerics in Excel
** Statistics/Numerical_Methods/Random Number Generation
* Time Series Analysis
* Multivariate Data Analysis
** Principal Component Analysis
** Factor Analysis for metrical data
** Factor Analysis for ordinal data
** Canonical Correlation Analysis
** Discriminant Analysis
* Analysis of Specific Datasets
** Analysis of Tuberculosis
* Appendix
** Authors
** Glossary
** Index
** Links
Combinatorics studies permutations and combinations of objects chosen from a sample space. A preliminary knowledge of combinatorics is necessary for a good command of statistics.

## Counting Principle

The Counting Principle is similar to the Multiplicative Principle. If a process involves $n$ steps and the $i$th step can be done in $x_{i}$ ways, then the entire process can be completed in $x_{1}\times x_{2}\times ...\times x_{n}$ different ways.

## Permutations

A permutation is a distinct arrangement of $n$ elements of a set. By the Counting Principle, the number of possible arrangements of $n$ objects in a set is $n\times (n-1)\times (n-2)\times ...\times 1=n!$. What if some of the elements are not distinct? Then, if there are $k$ distinct kinds of elements, the total number of possible arrangements is ${\frac {n!}{n_{1}!\times n_{2}!\times ...\times n_{k}!}}$. What if we are arranging the elements in a circle, rather than a line? Then the number of permutations is $(n-1)!$.
When faced with very large factorials, a useful approximation is Stirling's formula: $n!\approx {\sqrt {2\pi n}}({\frac {n}{e}})^{n}$
Now suppose we only choose r distinct elements from the set (without replacement). Then the number of possible permutations becomes ${\frac {n!}{(n-r)!}}=_{n}P_{r}$.

## Combinations

A combination is essentially a subset. It is like a permutation, except with no regard to order. Suppose we have a set of $n$ elements and take $r$ elements. The number of possible combinations is ${\frac {_{n}P_{r}}{r!}}={\frac {n!}{r!\times (n-r)!}}=_{n}C_{r}=\displaystyle {n \choose r}$.
Note also that $\sum _{r=0}^{k}\displaystyle {m \choose r}\times \displaystyle {n \choose k-r}=\displaystyle {m+n \choose k}$
Combinations are found in binomial expansion. Consider the following binomial expansions:
$(x+y)^{1}=x+y$
$(x+y)^{2}=x^{2}+2xy+y^{2}$
$(x+y)^{3}=x^{3}+3x^{2}y+3xy^{2}+y^{3}$
$(x+y)^{4}=x^{4}+4x^{3}y+6x^{2}y^{1}+4xy^{2}+y^{3}$
As you may have noticed from the above, for any positive integer $n$, $(x+y)^{n}=\sum _{i=0}^{n}[\displaystyle {n \choose r}\times x^{n-r}\times y^{r}]$
Another observation from the above is known as Pascal's law. It states that $\displaystyle {n \choose r}=\displaystyle {n-1 \choose r}+\displaystyle {n-1 \choose r-1}$
This allows us to construct Pascal's triangle, which is useful for determining combinations: