Giải tích 1 - Chapter 5: Discrete probability distributions

Chapter 5 DISCRETE PROBABILITY DISTRIBUTIONS Nguyen Tien Dung, MBA School of Economics and Management Website: https://sites.google.com/site/nguyentiendungbkhn Email: dung.nguyentien3@hust.edu.vn Main Contents 5.1 Random Variables 5.2 Developing Discrete Probability Distributions 5.3 Expected Value And Variance 5.4 Bivariate Distributions, Covariance, and Financial Portfolios 5.5 Binomial Probability Distribution 5.6 Poisson Probability Distribution 5.7 Hypergeometric Probability Distribution © Nguyễn Tiến Dũng Applied Statistics for Business 2 5.1 RANDOM VARIABLES ● A numerical description of the outcome of an experiment. ● A variable that assume random values, we don’t know in advance ● Example: ● The upper face of a dice ● The score of customer satisfaction in a survey ● Time between customer arrivals in minutes ● Two types: ● Discrete ● Continuous © Nguyễn Tiến Dũng Applied Statistics for Business 3 A discrete random variable ●A type of random variables that assume a finite number of values or an infinite sequence of discrete values © Nguyễn Tiến Dũng Applied Statistics for Business 4 Continuous Random Variables ●A random variable that may assume any numerical value in an interval or collection of intervals © Nguyễn Tiến Dũng Applied Statistics for Business 5 5.2 DISCRETE PROBABILITY DISTRIBUTIONS ● Required conditions for a discrete probability function: ● Example: the number of automobiles sold during a day at Dicarlo motors (Table 5.4) © Nguyễn Tiến Dũng Applied Statistics for Business 6 x f(x) 0 0.18 1 0.39 2 0.24 3 0.14 4 0.04 5 0.01 ( ) 0 ( ) 1 f x f x    Discrete Uniform Probability Function ● f(x) = 1/n ● n = the number of values the random variable may have ● Example: ● x = the number of dots on the upward face of a dice ● x = 1, 2, 3, 4, 5, 6 ● f(x) = 1/6 © Nguyễn Tiến Dũng Applied Statistics for Business 7 5.3 EXPECTED VALUE AND VARIANCE ●Expected value E(x) ● 𝐸 𝑥 = 𝜇 = 𝑥𝑓(𝑥) ● Example: Calculation of the expected value for the number of automobiles sold during a day at Dicarlo Motors ● Table 5.5 © Nguyễn Tiến Dũng Applied Statistics for Business 8 ●Variance ● 𝑉𝑎𝑟 𝑥 = 𝜎2 = 𝑥 − 𝜇 2𝑓(𝑥) © Nguyễn Tiến Dũng Applied Statistics for Business 9 5.4 Bivariate Distribution ●A probability distribution involving two random variables is called a bivariate probability distribution. © Nguyễn Tiến Dũng Applied Statistics for Business 10 A Bivariate Empirical Discrete Probability Distribution © Nguyễn Tiến Dũng Applied Statistics for Business 11 © Nguyễn Tiến Dũng Applied Statistics for Business 12 © Nguyễn Tiến Dũng Applied Statistics for Business 13 Covariance © Nguyễn Tiến Dũng Applied Statistics for Business 14 Correlation Coefficient © Nguyễn Tiến Dũng Applied Statistics for Business 15 Financial Applications © Nguyễn Tiến Dũng Applied Statistics for Business 16 E(ax+bx) and Var(ax+by) © Nguyễn Tiến Dũng Applied Statistics for Business 17 © Nguyễn Tiến Dũng Applied Statistics for Business 18 5.5 BINOMIAL PROBABILITY DISTRIBUTION ● Properties of a binomial experiment 1. The experiment consists of a sequence of n identical trials. 2. Two outcomes are possible on each trial. We refer to one outcome as a success and the other outcome as a failure. 3. The probability of a success, denoted by p, does not change from trial to trial. Consequently, the probability of a failure, denoted by 1 p, does not change from trial to trial. 4. The trials are independent. © Nguyễn Tiến Dũng Applied Statistics for Business 19 Martin Clothing Store Problem: The Tree Diagram © Nguyễn Tiến Dũng Applied Statistics for Business 20 Combination (n,x) © Nguyễn Tiến Dũng Applied Statistics for Business 21 © Nguyễn Tiến Dũng Applied Statistics for Business 22 © Nguyễn Tiến Dũng Applied Statistics for Business 23 ●Martin Clothing Store: Among the n = 3 customers, what is the probability of having x customers who make purchases? © Nguyễn Tiến Dũng Applied Statistics for Business 24 Graphical Representation Of The Probability Distribution For The Number Of Customers Making A Purchase © Nguyễn Tiến Dũng Applied Statistics for Business 25 Using Tables of Binomial Probabilities ● Appendix B © Nguyễn Tiến Dũng Applied Statistics for Business 26 Expected Value and Variance for the Binomial Distribution ● 𝐸 𝑥 = 𝜇 = 𝑛𝑝 ● 𝑉𝑎𝑟 𝑥 = 𝜎2 = 𝑛𝑝(1 − 𝑝) © Nguyễn Tiến Dũng Applied Statistics for Business 27 5.6 POISSON PROBABILITY DISTRIBUTION ● Properties of a 1. The probability of an occurrence is the same for any two intervals of equal length. 2. The occurrence or nonoccurrence in any interval is independent of the occurrence or nonoccurrence in any other interval. © Nguyễn Tiến Dũng Applied Statistics for Business 28 An Example Involving Time Intervals ● Suppose that we are interested in the number of arrivals at the drive-up teller window of a bank during a 15-minute period on weekday mornings. ● If we can assume that the probability of a car arriving is the same for any two time periods of equal length and that the arrival or non-arrival of a car in any time period is independent of the arrival or nonarrival in any other time period, the Poisson probability function is applicable. ● Suppose these assumptions are satisfied and an analysis of historical data shows that the average number of cars arriving in a 15-minute period of time is 10. © Nguyễn Tiến Dũng Applied Statistics for Business 29 ● The probability function: ● The probability of exactly 5 arrivals in 15 minutes © Nguyễn Tiến Dũng Applied Statistics for Business 30 An Example Involving Length or Distance Intervals ● Suppose we are concerned with the occurrence of major defects in a highway one month after resurfacing. ● We will assume that the probability of a defect is the same for any two highway intervals of equal length and that the occurrence or nonoccurrence of a defect in any one interval is independent of the occurrence or nonoccurrence of a defect in any other interval. ● Suppose we learn that major defects one month after resurfacing occur at the average rate of two per mile. (2 defects/mile) ● Let us find the probability of no major defects in a particular 3-mile section of the highway. ● Because we are interested in an interval with a length of 3 miles, μ = (2 defects/mile)(3 miles) = 6 represents the expected number of major defects over the 3-mile section of highway. © Nguyễn Tiến Dũng Applied Statistics for Business 31 Probability of no major defects over the 3-mile section of highway ( 0) ! x e f x x     5.7 HYPERGEOMETRIC PROBABILITY DISTRIBUTION ● Similar to binomial distribution ● Differences: ● Trials are not independent ● Probability of success changes from trial to trial © Nguyễn Tiến Dũng Applied Statistics for Business 32 Example ● Electric fuses produced by Ontario Electric are packaged in boxes of 12 units each. ● Suppose an inspector randomly selects 3 of the 12 fuses in a box for testing. If the box contains exactly 5 defective fuses, what is the probability that the inspector will find exactly 1 of the 3 fuses defective? © Nguyễn Tiến Dũng Applied Statistics for Business 33 Exercises for Homework Section Exercises 5.1 3, 4, 5, 6 5.2 9, 11, 13 5.3 17, 19, 20, 21 5.4 27 (add: to compute Var(x+y)), 28 5.5 33, 34, 35 5.6 45, 46, 47, 48 © Nguyễn Tiến Dũng Applied Statistics for Business 34