Stat 11 Lecture 12 Discrete Distributions (Chapter 4.3 & handout section 3.6) I. INTRODUCTION There are some kinds of random variables (called Bernoulli random variables) that can only take on two possible values. For example, a coin toss is a heads or tails situation and one could assign a value of "1" to heads and "0" to tails (if heads was of interest). In life, outcomes to certain random processes have yes/no, success/failure, up/down, male/female, buy/sell kind of outcomes. The probabilities associated with these Random Variables are π and 1-π. In other words, f(x) for a these Random Variables are: f(x) = P(X=x) = π if x=1 and 1-π if x=0. A. The Binomial Distribution (4.3) There are some situations when we are interested in a set or series of "1" or "0" outcomes. For example, you might be concerned with the number of defectives in the inspection of a sample of 100 computers. If you are in a situation when you have a fixed number of "things" or in this context "trials" and each of the trials is an independent random variable (independent Bernoulli RV) and each trial has the same probabilities of π and (1-π) this process can be described by a Binomial random variable. Definition Let S have a discrete distribution that has two outcomes: "success", with probability π, and "failure", with probability 1-π. 1. Theorem Consider a series of n independent trials. Suppose that each trial results in one of two possible outcomes, "success" or "failure". The probability of success "π" is the same from trial to trial. Let S denote the total number of successes in the n trials. Then S has a BINOMIAL DISTRIBUTION or a binomial probability distribution 2. Idea πs is the probability of exactly s independent successes in a row; )()1(sn−−π is the probability of exactly n-s independent failures in a row; (n choose s) counts the number of ways to arrange s successes and n-s failures. Written out on page 118 of your text, it is n!/s!(n-s)! )()1()()(snssnsSPsf−−⎟⎟⎠⎞⎜⎜⎝⎛===ππ will yield a discrete probability distribution. This probability distribution (the binomial) has certain properties. Two of them, the mean and the variance are very important for future work. )!(!!snsnsn−=⎟⎟⎠⎞⎜⎜⎝⎛where n! is n(n-1)(n-2)…(2)(1) and s!=s(s-1)(s-2)…(2)(1) this is called the binomial coefficient. It just tells you how many different ways there are to have “s” successes in N trials. Note: if s=n or if s=0 then the coefficient will be 1.0Stat 11 Lecture 12 Discrete Distributions (Chapter 4.3 & handout section 3.6) 3.Result: Probability Distribution An IRS auditor is going to randomly sample 1040 (long and short) tax returns in groups of 4 from the population of all tax returns. Suppose it is known that 58% of all tax returns contain errors. Please construct a probability distribution for the number of tax returns with errors in each sample of size 4. First construct a table of some type f(s) 0 1 2 3 4 P(S=s) Then fill it in. The probability function above just needs to be filled out for each possible s 0311.42.)58.1(58.04)0()0(4)4(0==−⎟⎟⎠⎞⎜⎜⎝⎛=== SPf 1719.)42)(.58)(.4()58.1(58.14)1()1(3)3(1==−⎟⎟⎠⎞⎜⎜⎝⎛=== SPf 3560.)42)(.58)(.6()58.1(58.24)2()2(22)2(2==−⎟⎟⎠⎞⎜⎜⎝⎛=== SPf 3278.)42)(.58)(.4()58.1(58.34)3()3(3)1(3==−⎟⎟⎠⎞⎜⎜⎝⎛=== SPf 1132.)58(.)58.1(58.44)4()4(4)0(4==−⎟⎟⎠⎞⎜⎜⎝⎛=== SPf check it: .0311+.1719+.3560+.3278+.1132=1.0 4. Result: Mean (Expectation) and Variance of a Binomial The formulas are the same as for any discrete random variable, that is: ∑≡≡allXiixpxXE )()(µ and ∑−=allXiixpx )()(22µσ but there are a couple of shorter formulas (page 122) πµnSE =≡)( and )1(2ππσ−= nS(the proof is on page 212 of your textbook) And remember the standard deviation is the square root of the variance. 5. Notes on the Binomial The key things to remember about a binomial – there are a number of trials, the trials are independent, and the probabilities are unchanging from trial to trial. The binomial is very commonly applied to model real random processes.Stat 11 Lecture 12 Discrete Distributions (Chapter 4.3 & handout section 3.6) The binomial distribution is characterized by two parameters, n and π. For each n and π, there exists a corresponding binomial distribution. Given a binomial random variable S, E(S) = nπ and var(S) = nπ (1-π). II. THE HISTOGRAM AGAIN (review pp 27-28, look ahead to Chapter 4.4) Recall the histogram. It looked like a bar charts in which the area of the bar is proportional to the number of observations having values in the range defining the bar. Histograms are generated for sample data but we can also construct histograms of populations. The population histogram describes the proportion of the population that lies between various limits (or numerical boundaries). It also describes the behavior of individual observations drawn at random from the population, that is, it gives the probability that an individual selected at random from the population will have a value between specified limits. Starting with chapter 4.4 it is important that you understand that population histograms describe the way individual observations behave. In the language of populations and probability, we don't say "population histogram". Instead, call it a probability density and/or a distribution function or a probability density function. When the area of a histogram is standardized to 1 or 100%, the histogram becomes a probability density function. The area of any portion of the histogram (the area under any part of the curve) is the proportion of the population in the designated region. It is also the probability that an individual selected at random will have a value in the designated region. Strictly speaking, a histogram is properly a density if tells you the proportion that lies between specified values. A (cumulative) distribution function is something else. It is a curve whose value is the proportion with values less than or equal to the value on the horizontal axis. Densities have the same name as their distribution functions. For example, later on, you will see that a bell-shaped curve is called a normal density. Observations that can be described by a normal density are said to follow a normal distribution. 0 .02 .04 .06 .08 .1 .12 .14 .16 .18