The Basics of Bayesian Statistics

Bayesian statistics mostly involves conditional probability, which is the the probability of an event A given event B, and it can be calculated using the Bayes rule. The concept of conditional probability is widely used in medical testing, in which false positives and false negatives may occur. A false positive can be defined as a positive outcome on…

Markov Chains

A popular model for systems that change over time in a random manner is theMarkov chain model. A Markov chain is a sequence of random variables, one foreach time. At each time, the corresponding random variable gives the state of thesystem. Also, the conditional distribution of each future state given the past statesand the present…

A/B testing

The A/B test (also known as a randomised controlled trial, or RCT, in the other sciences) is a powerful tool for product development. some motivations: With the rise of digital marketing led by tools including Google Analytics, Google Adwords, and Facebook Ads, a key competitive advantage for businesses is using A/B testing to determine effects…

Binomial Random Variables: Introduction

Binomial Random Variables So far, in our discussion about discrete random variables, we have been introduced to: The probability distribution, which tells us which values a variable takes, and how often it takes them. The mean of the random variable, which tells us the long-run average value that the random variable takes. The standard deviation…

Introduction to Normal Random Variables: Overview

In the Exploratory Data Analysis sections of this course, we encountered data sets, such as lengths of human pregnancies, whose distributions naturally followed a symmetric unimodal bell shape, bulging in the middle and tapering off at the ends. Many variables, such as pregnancy lengths, shoe sizes, foot lengths, and other human physical characteristics exhibit these properties: symmetry…

The Big Picture: Inference

Recall again the Big Picture, the four-step process that encompasses statistics: data production, exploratory data analysis, probability, and inference. We are about to start the fourth part of the process and the final section of this course, where we draw on principles learned in the other units (exploratory data analysis, producing data, and probability) in order…

Conditional Probability and Independence Introduction

Introduction In the last section, we established the five basic rules of probability, which include the two restricted versions of the Addition Rule and Multiplication Rule: The Addition Rule for Disjoint Events and the Multiplication Rule for Independent Events. We have also established a General Addition Rule for which the events need not be disjoint. In order to complete…

Probability Rules

Basic Probability Rules In the previous section we considered situations in which all the possible outcomes of a random experiment are equally likely, and learned a simple way to find the probability of any event in this special case. We are now moving on to learn how to find the probability of events in the…

How To Distribute Sample

Sampling Distributions Introduction Already on several occasions we have pointed out the important distinction between a population and a sample. In Exploratory Data Analysis, we learned to summarize and display values of a variable for a sample, such as displaying the blood types of 100 randomly chosen U.S. adults using a pie chart, or displaying the heights of 150…

Probability A short story

Sample Spaces As we saw in the previous section, probability questions arise when we are faced with a situation that involves uncertainty. Such a situation is called a random experiment, an experiment that produces an outcome that cannot be predicted in advance (hence the uncertainty). Here are a few examples of random experiments: Toss a coin…