Mean of Z = ( = (Variance of Z = (2 = (()(1-()The n Bernoulli trials are Z1 Z2 … Zn - Each Zi has possible values of 1 (“success”) or 0 (“failure”) - Pr [ Zi = 1 ] = ( and Pr [ Zi = 0 ] = (1-() for i=1, 2, …, nThe Binomial random variable is X = Z1 + Z2 + …+ Zn. X is distributed Binomial(n, π)PubHlth 540 – Fall 2011 4. Bernoulli and Binomial Page 1 of 21 Unit 4 The Bernoulli and Binomial Distributions “If you believe in miracles, head for the Keno lounge” - Jimmy the Greek The Amherst Regional High School provides flu vaccinations to a random sample of 200 students. How many will develop the flu? A new treatment for stage IV melanoma is given to 75 cases. How many will survive two or more years? In a sample of 300 cases of uterine cancer, how many have a history of IUD use? The number of “events” in each of these scenarios is a random variable that is modeled well using a Binomial probability distribution. When the number of trials is just one, the probability model is called a Bernoulli trial. The Bernoulli and Binomial probability distributions are used to describe the chance occurrence of “success/failure” outcomes. They are also the basis of logistic regression which is used to identify the possibly multiple predictors of “success/failure” outcomes. Nature Population/ Sample Observation/ Data Relationships/ Modeling Analysis/ SynthesisPubHlth 540 – Fall 2011 4. Bernoulli and Binomial Page 2 of 21 Table of Contents Topic 1. Unit Roadmap …………………………………………………. 2. Learning Objectives ……………………………………………. 3. Introduction to Discrete Probability Distributions ……………... 4. Statistical Expectation ……………………………………….…. 5. The Population Variance is a Statistical Expectation ………….. 6. The Bernoulli Distribution …………………..………………… 7. Introduction to Factorials and Combinatorials …………….…… 8. The Binomial Distribution ………………………………….…. 9. Calculation of Binomial Probabilities …….……………….…… 10. Resources for the Binomial Distribution …………………..…… 3457101113161921 Nature Population/ Sample Observation/ Data Relationships/ Modeling Analysis/ SynthesisPubHlth 540 – Fall 2011 4. Bernoulli and Binomial Page 3 of 21 1. Unit Roadmap Nature/ Populations Unit 4. Bernoulli and Binomial Sample Data are the measurements of our observations of nature. This unit focuses on nominal data that are binary. Previously, we learned that data can be of several types – nominal, ordinal, quantitative, etc. A nominal variable that is binary or dichotomous has exactly two possible values. Examples are vital status (alive/dead), exposure (yes/no), tumor remission(yes/no), etc. The “frequentist” view of probability says that probability is the relative frequency in an indefinitely large number of trials. In this framework, a probability distribution model is a model of chance. It describes the way that probability is distributed among the possible values that a random variable can take on. The Bernoulli and Binomial probability distribution models are often very good descriptions of patterns of occurrence of events that are of interest in public health; eg - mortality, disease, and exposure. Observation/ Data Relationships Modeling Analysis/ Synthesis Nature Population/ Sample Observation/ Data Relationships/ Modeling Analysis/ SynthesisPubHlth 540 – Fall 2011 4. Bernoulli and Binomial Page 4 of 21 2. Learning Objectives When you have finished this unit, you should be able to:  Explain the “frequentist” approach to probability.  Define a discrete probability distribution.  Explain statistical expectation for a discrete random variable..  Define the Bernoulli probability distribution model.  Explain how to “count the # ways” using the tools of factorials and combinatorials.  Define the Binomial probability distribution model.  Calculate binomial probabilities. Nature Population/ Sample Observation/ Data Relationships/ Modeling Analysis/ SynthesisPubHlth 540 – Fall 2011 4. Bernoulli and Binomial Page 5 of 21 3. Introduction to Discrete Probability Distributions A discrete probability distribution is defined by (i) a listing of all the possible random variable values, together with (ii) their associated probabilities of occurrence. • The listing of possible random variable outcomes must comprise ALL the possibilities (be exhaustive) • Each possibility has a likelihood of occurrence (“chances of occurrence”) that is a number somewhere between 0 and 1. • Looking ahead … We’ll have to refine these notions when we come to speaking about continuous distributions because, in those situations, the number of possible outcomes is infinite!. Example: Gender of a randomly selected student • We’ll use capital X as our placeholder for the random variable name: X = Gender of randomly selected student from the population of students at a University • We’ll use small x as our placeholder for a value of the random variable X: x = 0 if gender of the selected student is male x = 1 if gender of the selected student is female Nature Population/ Sample Observation/ Data Relationships/ Modeling Analysis/ SynthesisPubHlth 540 – Fall 2011 4. Bernoulli and Binomial Page 6 of 21 Value of the Random Variable X is x = Probability that X has value x is Pr [ X = x ] = 0 = male 1 = female Note that this roster exhausts all possibilities. 0.53 0.47 Note that the sum of these individual probabilities, because the sum is taken over all possibilities, is 100% or 1.00. Some useful terminology - 1. For discrete random variables, a probability model is the set of assumptions