# Probability and statistics EBook

SOCR Books: This is a General Statistics Curriculum EBook, which includes Advanced-Placement (AP) probability and statistics materials.

## Contents

- 1 Preface
- 2 Chapter I: Introduction to Statistics
- 3 Chapter II: Describing, Exploring, and Comparing Data
- 4 Chapter III: Probability
- 5 Chapter IV: Probability Distributions
- 6 Chapter V: Normal Probability Distribution
- 7 Chapter VI: Relations Between Distributions
- 8 Chapter VII: Point and Interval Estimates
- 9 Chapter VIII: Hypothesis Testing
- 10 Chapter IX: Inferences From Two Samples
- 11 Chapter X: Correlation and Regression
- 12 Chapter XI: Analysis of Variance (ANOVA)
- 13 Chapter XII: Non-Parametric Inference
- 13.1 Differences of Medians (Centers) of Two Paired Samples
- 13.2 Differences of Medians (Centers) of Two Independent Samples
- 13.3 Differences of Proportions of Two Samples
- 13.4 Differences of Medians of Several Independent Samples
- 13.5 Differences of Variances of Independent Samples (Variance Homogeneity)

- 14 Chapter XIII: Multinomial Experiments and Contingency Tables
- 15 Chapter XIV:Bayesian Statistics
- 16 Chapter XV: Other Common Continuous Distributions
- 16.1 Gamma Distribution
- 16.2 Exponential Distribution
- 16.3 Pareto Distribution
- 16.4 Beta Distribution
- 16.5 Laplace (Double Exponential) Distribution
- 16.6 Cauchy Distribution
- 16.7 Chi-Square Distribution
- 16.8 Fisher's F Distribution
- 16.9 Johnson SB Distribution
- 16.10 Rice Distribution
- 16.11 Uniform Distribution

- 17 Additional EBook Chapters (under Development)

## Preface

This is an Internet-based *probability and statistics E-Book*. The materials, tools and demonstrations presented in this E-Book would be very useful for advanced-placement (AP) statistics educational curriculum. The E-Book is initially developed by the UCLA Statistics Online Computational Resource (SOCR). However, all statistics instructors, researchers and educators are encouraged to contribute to this project and improve the content of these learning materials.

There are 4 novel features of this specific *Statistics EBook*. It is community-built, completely open-access (in terms of use and contributions), blends information technology, scientific techniques and modern pedagogical concepts, and is multilingual.

### Format

Follow the instructions in this page to expand, revise or improve the materials in this E-Book.

### Learning and Instructional Usage

This section describes the means of traversing, searching, discovering and utilizing the SOCR Statistics EBook resources in both formal and informal learning setting. The problems of each section in the E-Book are shown here.

### Copyrights

The Probability and Statistics EBook is a freely and openly accessible electronic book developed by SOCR and the general community.

## Chapter I: Introduction to Statistics

### The Nature of Data and Variation

Although natural phenomena in the real life are unpredictable, the designs of experiments are bounded to generate data that varies because of intrinsic (internal to the system) or extrinsic (due to the ambient environment) effects. How many natural processes or phenomena in the real life that have an exact mathematical closed-form description and are completely deterministic can we describe? How do we model the rest of the processes that are unpredictable and have random characteristics?

### Uses and Abuses of Statistics

Statistics is the science of variation, randomness and chance. As such, statistics is different from other sciences, where the processes being studied obey exact deterministic mathematical laws. Statistics provides quantitative inference represented as long-time probability values, confidence or prediction intervals, odds, chances, etc., which may ultimately be subjected to varyious interpretations. The phrase *Uses and Abuses of Statistics* refers to the notion that in some cases statistical results may be used as evidence to seemingly opposite theses. However, most of the time, common principles of logic allow us to disambiguate the obtained statistical inference.

### Design of Experiments

Design of experiments is the blueprint for planning a study or experiment, performing the data collection protocol and controlling the study parameters for accuracy and consistency. Data, or information, is typically collected in regard to a specific process or phenomenon being studied to investigate the effects of some controlled variables (independent variables or predictors) on other observed measurements (responses or dependent variables). Both types of variables are associated with specific observational units (living beings, components, objects, materials, etc.)

### Statistics with Tools (Calculators and Computers)

All methods for data analysis, understanding or visualizing are based on models that often have compact analytical representations (e.g., formulas, symbolic equations, etc.) Models are used to study processes theoretically. Empirical validations of the utility of models are achieved by inputting data and executing tests of the models. This validation step may be done manually, by computing the model prediction or model inference from recorded measurements. This process may be possibly done by hand, but only for small numbers of observations (<10). In practice, we write (or use existent) algorithms and computer programs that automate these calculations for greater efficiency, accuracy and consistency in applying models to larger datasets.

## Chapter II: Describing, Exploring, and Comparing Data

### Types of Data

There are two important concepts in any data analysis - **Population** and **Sample**.
Each of these may generate data of two major types - **Quantitative** or **Qualitative** measurements.

### Summarizing Data with Frequency Tables

There are two important ways to describe a data set (sample from a population) - **Graphs** or **Tables**.

### Pictures of Data

There are many different ways to display and graphically visualize data. These graphical techniques facilitate the understanding of the dataset and enable the selection of an appropriate statistical methodology for the analysis of the data.

### Measures of Central Tendency

There are three main features of populations (or sample data) that are always critical in understanding and interpreting their distributions - **Center**, **Spread** and **Shape**. The main measures of centrality are **Mean**, **Median** and **Mode(s)**.

### Measures of Variation

There are many measures of (population or sample) spread, e.g., the range, the variance, the standard deviation, mean absolute deviation, etc. These are used to assess the dispersion or variation in the population.

### Measures of Shape

The **shape** of a distribution can usually be determined by looking at a histogram of a (representative) sample from that population; Frequency Plots, Dot Plots or Stem and Leaf Displays may be helpful.

### Statistics

Variables can be summarized using statistics - functions of data samples.

### Graphs and Exploratory Data Analysis

Graphical visualization and interrogation of data are critical components of any reliable method for statistical modeling, analysis and interpretation of data.

## Chapter III: Probability

Probability is important in many studies and discipline because measurements, observations and findings are often influenced by variation. In addition, probability theory provides the theoretical groundwork for statistical inference.

### Fundamentals

Some fundamental concepts of probability theory include random events, sampling, types of probabilities, event manipulations and axioms of probability.

### Rules for Computing Probabilities

There are many important rules for computing probabilities of composite events. These include conditional probability, statistical independence, multiplication and addition rules, the law of total probability and the Bayesian Rule.

### Probabilities Through Simulations

Many experimental setting require probability computations of complex events. Such calculations may be carried out exactly, using theoretical models, or approximately, using estimation or simulations.

### Counting

There are many useful counting principles (including permutations and combinations) to compute the number of ways that certain arrangements of objects can be formed. This allows counting-based estimation of complex events' probabilities.

## Chapter IV: Probability Distributions

There are two basic types of processes that we observe in nature - **Discrete** and **Continuous**. We begin by discussing several important discrete random processes, emphasizing the different distributions, expectations, variances and applications. In the next chapter, we will discuss their continuous counterparts and other continuous distributions are discussed in a later chapter. The complete list of all SOCR Distributions is available here.

### Random Variables

To simplify the calculations of probabilities, we will define the concept of a **random variable** which will allow us to study uniformly various processes with the same mathematical and computational techniques.

### Expectation (Mean) and Variance

The expectation and the variance for any discrete random variable or process are important measures of Centrality and Dispersion. This section also presents the definitions of some common population- or sample-based moments.

### Bernoulli and Binomial Experiments

The **Bernoulli** and **Binomial** processes provide the simplest models for discrete random experiments.

### Multinomial Experiments

**Multinomial processes** extend the Binomial experiments for the situation of multiple possible outcomes.

### Geometric, Hypergeometric, Negative Binomial and Negative Multinomial

The **Geometric, Hypergeometric, Negative Binomial, and Negative Multinomial distributions** provide computational models for calculating probabilities for a large number of experiment and random variables. This section presents the theoretical foundations and the applications of each of these discrete distributions.

### Poisson Distribution

The **Poisson distribution** models many different discrete processes where the probability of the observed phenomenon is constant in time or space. Poisson distribution may be used as an approximation to the Binomial distribution.

## Chapter V: Normal Probability Distribution

The Normal Distribution is perhaps the most important model for studying quantitative phenomena in the natural and behavioral sciences - this is due to the Central Limit Theorem. Many numerical measurements (e.g., weight, time, etc.) can be well approximated by the normal distribution. Other commonly used continuous distributions are discussed in a later chapter.

### The Standard Normal Distribution

The Standard Normal Distribution is the simplest version (zero-mean, unit-standard-deviation) of the (General) Normal Distribution. Yet, it is perhaps the most frequently used version because many tables and computational resources are explicitly available for calculating probabilities.

### Nonstandard Normal Distribution: Finding Probabilities

In practice, the mechanisms underlying natural phenomena may be unknown, yet the use of the normal model can be theoretically justified in many situations to compute critical and probability values for various processes.

### Nonstandard Normal Distribution: Finding Scores (Critical Values)

In addition to being able to compute probability (p) values, we often need to estimate the critical values of the Normal Distribution for a given p-value.

### Multivariate Normal Distribution

The multivariate normal distribution (also known as multivariate Gaussian distribution) is a generalization of the univariate (one-dimensional) normal distribution to higher dimensions (2D, 3D, etc.) The multivariate normal distribution is useful in studies of correlated real-valued random variables.

## Chapter VI: Relations Between Distributions

In this chapter, we will explore the relationships between different distributions. This knowledge will help us to compute difficult probabilities using reasonable approximations and identify appropriate probability models, graphical and statistical analysis tools for data interpretation. The complete list of all SOCR Distributions is available here and the Probability Distributome project provides an interactive graphical interface for exploring the relations between different distributions.

### The Central Limit Theorem

The exploration of the relations between different distributions begins with the study of the **sampling distribution of the sample average**. This will demonstrate the universally important role of normal distribution.

### Law of Large Numbers

Suppose the relative frequency of occurrence of one event whose probability to be observed at each experiment is *p*. If we repeat the same experiment over and over, the ratio of the observed frequency of that event to the total number of repetitions converges towards *p* as the number of experiments increases. Why is that and why is this important?

### Normal Distribution as Approximation to Binomial Distribution

Normal Distribution provides a valuable approximation to Binomial when the sample sizes are large and the probability of successes and failures is not close to zero.

### Poisson Approximation to Binomial Distribution

Poisson provides an approximation to Binomial Distribution when the sample sizes are large and the probability of successes or failures is close to zero.

### Binomial Approximation to Hypergeometric

Binomial Distribution is much simpler to compute, compared to Hypergeometric, and can be used as an approximation when the population sizes are large (relative to the sample size) and the probability of successes is not close to zero.

### Normal Approximation to Poisson

The Poisson can be approximated fairly well by Normal Distribution when λ is large.

## Chapter VII: Point and Interval Estimates

Estimation of population parameters is critical in many applications. Estimation is most frequently carried in terms of point-estimates or interval (range) estimates for population parameters that are of interest.

### Method of Moments and Maximum Likelihood Estimation

There are many ways to obtain point (value) estimates of various population parameters of interest, using observed data from the specific process we study. The **method of moments** and the **maximum likelihood estimation** are among the most popular ones frequently used in practice.

### Estimating a Population Mean: Large Samples

This section discusses how to find point and interval estimates when the sample-sizes are large.

### Estimating a Population Mean: Small Samples

Next, we discuss point and interval estimates when the sample-sizes are small. Naturally, the point estimates are less precise and the interval estimates produce wider intervals, compared to the case of large-samples.

### Student's T distribution

The **Student's T-Distribution** arises in the problem of estimating the mean of a normally distributed population when the sample size is small and the population variance is unknown.

### Estimating a Population Proportion

**Normal Distribution** is an appropriate model for proportions, when the sample size is large enough. In this section, we demonstrate how to obtain point and interval estimates for population proportion.

### Estimating a Population Variance

In many processes and experiments, controlling the amount of variance is of critical importance. Thus the ability to assess variation, using point and interval estimates, facilitates our ability to make inference, revise manufacturing protocols, improve clinical trials, etc.

### Confidence Intervals Activity

This activity demonstrates the usage and functionality of SOCR General Confidence Interval Applet. This applet is complementary to the SOCR Simple Confidence Interval Applet and its corresponding activity.

## Chapter VIII: Hypothesis Testing

**Hypothesis Testing** is a statistical technique for decision making regarding populations or processes based on experimental data. It quantitatively answers the possibility that chance alone might be responsible for the observed discrepancies between a theoretical model and the empirical observations.

### Fundamentals of Hypothesis Testing

In this section, we define the core terminology necessary to discuss Hypothesis Testing (Null and Alternative Hypotheses, Type I and II errors, Sensitivity, Specificity, Statistical Power, etc.)

### Testing a Claim about a Mean: Large Samples

As we already saw how to construct point and interval estimates for the population mean in the large sample case, we now show how to do hypothesis testing in the same situation.

### Testing a Claim about a Mean: Small Samples

We continue with the discussion on inference for the population mean of small samples.

### Testing a Claim about a Proportion

When the sample size is large, the sampling distribution of the sample proportion \(\hat{p}\) is approximately Normal, by CLT. This helps us formulate hypothesis testing protocols and compute the appropriate statistics and p-values to assess significance.

### Testing a Claim about a Standard Deviation or Variance

The significance testing for the variation or the standard deviation of a process, a natural phenomenon or an experiment is of paramount importance in many fields. This chapter provides the details for formulating testable hypotheses, computation, and inference on assessing variation.

## Chapter IX: Inferences From Two Samples

In this chapter, we continue our pursuit and study of significance testing in the case of having two populations. This expands the possible applications of one-sample hypothesis testing we saw in the previous chapter.

### Inferences About Two Means: Dependent Samples

We need to clearly identify whether samples we compare are **Dependent** or **Independent** in all study designs. In this section, we discuss one specific dependent-samples case - **Paired Samples**.

### Inferences About Two Means: Independent Samples

**Independent** Samples designs refer to experiments or observations where all measurements are individually independent from each other within their groups and the groups are independent. In this section, we discuss inference based on independent samples.

### Comparing Two Variances

In this section, we compare **variances (or standard deviations)** of two populations using randomly sampled data.

### Inferences about Two Proportions

This section presents the **significance testing** and **inference on equality** of proportions from two independent populations.

## Chapter X: Correlation and Regression

Many scientific applications involve the analysis of relationships between two or more variables involved in a process of interest. We begin with the simplest of all situations where **Bivariate Data** (X and Y) are measured for a process and we are interested in determining the association, relation or an appropriate model for these observations (e.g., fitting a straight line to the pairs of (X,Y) data).

### Correlation

The **Correlation** between X and Y represents the first bivariate model of association which may be used to make predictions.

### Regression

We are now ready to discuss the modeling of linear relations between two variables using **Regression Analysis**. This section demonstrates this methodology for the SOCR California Earthquake dataset.

### Variation and Prediction Intervals

In this section, we discuss point and interval estimates about the slope of linear models.

### Multiple Regression

Now, we are interested in determining linear regressions and multilinear models of the relationships between one dependent variable Y and many independent variables \(X_i\).

## Chapter XI: Analysis of Variance (ANOVA)

### One-Way ANOVA

We now expand our inference methods to study and compare *k* **independent** samples. In this case, we will be decomposing the entire variation in the data into independent components.

### Two-Way ANOVA

Now we focus on decomposing the variance of a dataset into (independent/orthogonal) components when we have two (grouping) factors. This procedure called **Two-Way Analysis of Variance**.

## Chapter XII: Non-Parametric Inference

To be valid, many statistical methods impose (parametric) requirements about the format, parameters and distributions of the data to be analyzed. For instance, the Independent T-Test requires the distributions of the two samples to be Normal, whereas Non-Parametric (distribution-free) statistical methods are often useful in practice, and are less-powerful.

### Differences of Medians (Centers) of Two Paired Samples

The **Sign Test** and the **Wilcoxon Signed Rank Test** are the simplest non-parametric tests which are also alternatives to the One-Sample and Paired T-Test. These tests are applicable for paired designs where the data is not required to be normally distributed.

### Differences of Medians (Centers) of Two Independent Samples

The **Wilcoxon-Mann-Whitney (WMW) Test** (also known as Mann-Whitney U Test, Mann-Whitney-Wilcoxon Test, or Wilcoxon rank-sum Test) is a *non-parametric* test for assessing whether two samples come from the same distribution.

### Differences of Proportions of Two Samples

Depending upon whether the samples are dependent or independent, we use different statistical tests.

### Differences of Medians of Several Independent Samples

We now extend the multi-sample inference which we discussed in the ANOVA section, to the situation where the ANOVA assumptions are invalid.

### Differences of Variances of Independent Samples (Variance Homogeneity)

There are several tests for variance equality in *k* samples. These tests are commonly known as tests for **Homogeneity of Variances**.

## Chapter XIII: Multinomial Experiments and Contingency Tables

### Multinomial Experiments: Goodness-of-Fit

The **Chi-Square Test** is used to test if a data sample comes from a population with specific characteristics.

### Contingency Tables: Independence and Homogeneity

The **Chi-Square Test** may also be used to test for independence (or association) between two variables.

## Chapter XIV:Bayesian Statistics

### Preliminaries

This section will establish the groundwork for Bayesian Statistics. Probability, Random Variables, Means, Variances, and the Bayes’ Theorem will all be discussed.

### Bayesian Inference for the Normal Distribution

In this section, we will provide the basic framework for Bayesian statistical inference. Generally, we take some prior beliefs about some hypothesis and then modify these prior beliefs, based on some data that we collect, in order to arrive at posterior beliefs. Another way to think about Bayesian Inference is that we are using new evidence or observations to update the probability that a hypothesis is true.

### Some Other Common Distributions

This section explains the binomial, Poisson, and uniform distributions in terms of Bayesian Inference (also see the chapter on other common distributions).

### Hypothesis Testing

This section will talk about both the classical approach to hypothesis testing and also the Bayesian approach.

### Two Sample Problems

This section discusses two sample problems, with variances unknown, both equal and unequal. The Behrens-Fisher controversy is also discussed.

### Hierarchical Models

Hierarchical linear models are statistical models of parameters that vary at more than a level. These models are seen as generalizations of linear models and may extend to non-linear models. Any underlying correlations in the particular model must be represented in analysis for correct inference to be drawn.

### The Gibbs Sampler and Other Numerical Methods

Topics covered will include Monte Carlo Methods, Markov Chains, the EM Algorithm, and the Gibbs Sampler.

## Chapter XV: Other Common Continuous Distributions

Earlier we discussed some classes of commonly used Discrete and Continuous distributions. Below are some continuous distributions with broad range of applications. The complete list of all SOCR Distributions is available here. The Probability Distributome Project provides an interactive navigator for traversal, discovery and exploration of probability distribution properties and interrelations.

### Gamma Distribution

The Gamma distribution is a distribution that arises naturally in processes for which the waiting times between events are relevant. It can be thought of as a waiting time between Poisson distributed events.

### Exponential Distribution

The Exponential distribution is a special case of the Gamma distribution. Whereas the Gamma distribution is the waiting time for more than one event, the Exponential distribution describes the time between a single Poisson event.

### Pareto Distribution

The Pareto distribution is a skewed, heavy-tailed distribution that is sometimes used to model the distribution of incomes. The basis of the distribution is that a high proportion of a population has low income while only a few people have very high incomes.

### Beta Distribution

The Beta distribution is a distribution that models events which are constrained to take place within an interval defined by a minimum and maximum value.

### Laplace (Double Exponential) Distribution

The Laplace distribution is a distribution that is symmetrical and more “peaky” than a Normal distribution. The dispersion of the data around the mean is higher than that of a Normal distribution. Laplace distribution is also sometimes called the Double Exponential distribution.

### Cauchy Distribution

The Cauchy distribution, also called the Lorentzian distribution or Lorentz distribution, is a continuous distribution describing resonance behavior.

### Chi-Square Distribution

The Chi-Square distribution is used in the chi-square tests for goodness of fit of an observed distribution to a theoretical one and the independence of two criteria of classification of qualitative data. It is also used in confidence interval estimation for a population standard deviation of a normal distribution from a sample standard deviation. The Chi-Square distribution is a special case of the Gamma distribution.

### Fisher's F Distribution

Commonly used as the null distribution of a test statistic, such as in analysis of variance (ANOVA).

### Johnson SB Distribution

The Johnson SB distribution is related to the Normal distribution. Four parameters are needed\[\gamma, \delta, \lambda, \epsilon.\] It is a continuous distribution defined on bounded range \( \epsilon \leq x \leq \epsilon + \lambda \), and the distribution can be symmetric or asymmetric.

### Rice Distribution

Also known as the Rician Distribution, the Rice distribution is the probability distribution of the absolute value of a circular bivariate normal random variable with potentially non-zero mean.

### Uniform Distribution

In the Continuous Uniform distribution, all intervals of the same length are equally probable. In the Discrete Uniform distribution, there are n equally spaced values, each of which have the same 1/n probability of being observed.

## Additional EBook Chapters (under Development)

- SOCR Home page: http://www.socr.ucla.edu

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