Probability Introduction Shifting our focus We were studying statistics data displays sampling The next few lectures focus on probability randomness Why What is Probability Probability is used to quantify uncertainty about the future Probability is a way to model randomness Why Study Probability in a Statistics Course Probability is the language with which we express and interpret assessment of uncertainty in a formal statistical analysis We take a random sample from a population Board Statistical Cartoon random sample 3 2 Random Sampling Random Sampling Most of the formal methods of statistical inference we will use in this class are based on the assumption that the individual units in the sample are sampled at random from the population of interest Ignore for the present that in practice individuals are almost never sampled at random in a very formal sense from the population of interest Simple Random Sampling Taking a simple random sample of size n is equivalent to the process of 1 representing every individual from a population with a single ticket 2 putting the tickets into large box 3 mixing the tickets thoroughly 4 drawing out n tickets without replacement Simple Random Sampling The defining characteristic of the process of simple random sampling is that every possible sample of size n has the same chance of being selected In particular this means that a every individual has the same chance of being included in the sample and that b members of the sample are chosen independently of each other Other Random Sampling Strategies Stratified random sampling and cluster sampling are examples of random sampling processes that are not simple Data analysis for these types of sampling strategies go beyond the scope of this course Inference from Samples to Populations Statistical inference involves making statements about populations on the basis of analysis of sampled data The Simple random sampling model is useful because it allows precise mathematical description of the random distribution of the discrepancy between statistical estimates and population parameters This is known as chance error due to random sampling When using the random sampling model it is important to ask what is the population to which the results will be generalized The use statistical methods that assume random sampling on data that is not collected as a random sample is prone to sampling bias in which individuals do not have the same chance of being sampled Sampling bias can lead to incorrect statistical inferences because the sample is unrepresentative of the population in important ways Examples For you You should read the following Examples p 75 76 Example 3 2 Example 3 7 Further examples of sampling procedures and sampling bias 3 3 Introduction to Probability Outcome Space Definition The outcome space is the set of possible simple outcomes from a random experiment often denoted by Example In a single die roll the set of possible outcomes is 1 2 3 4 5 6 Events Definition An event is a set of possible outcomes Example In a single die roll possible events include A the die roll is even B the die roll is a 6 C the die roll is 4 or less Probability Definition The probability of an event E denoted P E is a numerical measure between 0 and 1 that represents the likelihood of the event E in some probability model Probabilities assigned to events must follow a number of rules Probability Definition The probability of an event E denoted P E is a numerical measure between 0 and 1 that represents the likelihood of the event E in some probability model Probabilities assigned to events must follow a number of rules Note 1 Probability is a numerical measure of the likelihood of an event 2 Probabilities are always between 0 and 1 inclusive 3 Notation The probability of an event E is written P E Probability Definition The probability of an event E denoted P E is a numerical measure between 0 and 1 that represents the likelihood of the event E in some probability model Probabilities assigned to events must follow a number of rules Note 1 Probability is a numerical measure of the likelihood of an event 2 Probabilities are always between 0 and 1 inclusive 3 Notation The probability of an event E is written P E Example The probability P the die roll is a 6 equals 1 6 under a probability model that gives equal probability to each possible result but could be different under a different model Examples 1 If a fair coin is tossed the probability of a head is P Heads 0 5 2 If bucket contains 34 white balls and 66 red balls and a ball is drawn at random the probability that the drawn ball is white is P white 34 100 0 34 Probability Rules Non negativity For any event E 0 P E 1 Outcome space The probability of the event of all possible outcomes is 1 Complements P E c 1 P E 3 7 Random Variables Introduction to Random Variables Definition A random variable is a variable that depends on an outcome of a chance random experiment Examples Variable versus a Random Variable Examples 1 Bret s height Examples Variable versus a Random Variable Examples 1 Bret s height not random Examples Variable versus a Random Variable Examples 1 2 Bret s height not random Select a student in class at random and record the student s height Examples Variable versus a Random Variable Examples 1 2 Bret s height not random Select a student in class at random and record the student s height random Examples Variable versus a Random Variable Examples 1 2 3 Bret s height not random Select a student in class at random and record the student s height random The total height of everyone in 331 SMI right now Examples Variable versus a Random Variable Examples 1 2 3 Bret s height not random Select a student in class at random and record the student s height random The total height of everyone in 331 SMI right now not random Examples Variable versus a Random Variable Examples 1 2 3 4 Bret s height not random Select a student in class at random and record the student s height random The total height of everyone in 331 SMI right now not random Role a die and then record the number on its face Examples Variable versus a Random Variable Examples 1 2 3 4 Bret s height not random Select a student in class at random and record the student s height random The total height of everyone in 331 SMI right now not random Role a die and then record the number on its face random Random Variables A random variable is a rule that attaches a numerical value to a chance outcome Associated
View Full Document
Unlocking...