Lecture Business statistics in practice (7/e): Chapter 5 - Bowerman, O'Connell, Murphree

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Chapter 5 - Discrete random variables. After mastering the material in this chapter, you will be able to: Explain the difference between a discrete random variable and a continuous random variable, find a discrete probability distribution and compute its mean and standard deviation, use the binomial distribution to compute probabilities,...

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Nội dung Text: Lecture Business statistics in practice (7/e): Chapter 5 - Bowerman, O'Connell, Murphree

Chapter 5 Discrete Random Variables McGrawHill/Irwin Copyright © 2014 by The McGrawHill Companies, Inc. All rights reserved.
Discrete Random Variables 5.1 Two Types of Random Variables 5.2 Discrete Probability Distributions 5.3 The Binomial Distribution 5.4 The Poisson Distribution (Optional) 5.5 The Hypergeometric Distribution (Optional) 5.6 Joint Distributions and the Covariance (Optional) 52
LO5-1: Explain the difference between a discrete random 5.1 Two Types of Random Variables variable and a continuous random variable.  Random variable: a variable that assumes numerical values that are determined by the outcome of an experiment ◦ Discrete ◦ Continuous  Discrete random variable: Possible values can be counted or listed ◦ The number of defective units in a batch of 20 ◦ A listener rating (on a scale of 1 to 5) in an AccuRating music survey  Continuous random variable: May assume any numerical value in one or more intervals ◦ The waiting time for a credit card authorization ◦ The interest rate charged on a business loan 53
LO5-2: Find a discrete probability distribution and compute its mean 5.2 Discrete Probability Distributions and standard deviation. The probability distribution of a discrete random variable is a table, graph or formula that gives the probability associated with each possible value that the variable can assume Notation: Denote the values of the random variable by x and the value’s associated probability by p(x) 54
LO5-2 Discrete Probability Distribution Properties 1. For any value x of the random variable, p(x) 0 2. The probabilities of all the events in the sample space must sum to 1, that is… px 1 all x 55
LO5-3: Use the binomial distribution to compute probabilities. 5.3 The Binomial Distribution  The binomial experiment characteristics… 1. Experiment consists of n identical trials 2. Each trial results in either “success” or “failure” 3. Probability of success, p, is constant from trial to trial  The probability of failure, q, is 1 – p 1. Trials are independent  If x is the total number of successes in n trials of a binomial experiment, then x is a binomial random variable 56
LO5-3 Binomial Distribution Continued  For a binomial random variable x, the probability of x successes in n trials is given by the binomial distribution: n! x n x p x = p q x! n x !  n! is read as “n factorial” and n! = n × (n1) × (n2) × ... × 1  0! =1  Not defined for negative numbers or fractions 57
LO5-4: Use the Poisson distribution to compute probabilities (Optional). 5.4 The Poisson Distribution  Consider the number of times an event occurs over an interval of time or space, and assume that 1. The probability of occurrence is the same for any intervals of equal length 2. The occurrence in any interval is independent of an occurrence in any nonoverlapping interval  If x = the number of occurrences in a specified interval, then x is a Poisson random variable 58
LO5-5: Use the hypergeometric distribution to compute probabilities (Optional). 5.5 The Hypergometric Distribution (Optional) Population consists of N items ◦r of these are successes ◦(Nr) are failures If we randomly select n items without replacement, the probability that x of the n items will be successes is given by the hypergeometric probability formula r N r x n x P ( x) N n 59
LO5-5 The Mean and Variance of a Hypergeometric Random Variable Mean r x n N Variance 2 r r N n x n 1 N N N 1 510
LO5-6: Compute and understand the covariance between two random variables (Optional). 5.6 Joint Distributions and the Covariance (Optional) 511
LO5-6 Four Properties of Expected Values and Variances 1. If a is a constant and x is a random variable, then μax=aμx 2. If x1,x2,…,xn are random variables, then μ(x1,x2,…,xn)= μx1 + μx2 + … + μxn 3. If a is a constant and x is a random variable, then σ2ax=a2σ2x 4. If x1,x2,…,xn are statistically independent random variables, then the covariance is zero ◦ Also, σ2(x1,x2,…,xn)= σ2x1+ σ2x2+…+ σ2xn 512