Chapter 5 Discrete Random Variables

Chapter 5Discrete Random VariablesCopyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/IrwinDiscrete Random Variables5.1 Two Types of Random Variables5.2 Discrete Probability Distributions5.3 The Binomial Distribution5.4 The Poisson Distribution (Optional)5.5 The Hypergeometric Distribution (Optional)5.6 Joint Distributions and the Covariance (Optional)5-*5.1 Two Types of Random VariablesRandom variable: a variable that assumes numerical values that are determined by the outcome of an experimentDiscreteContinuousDiscrete random variable: Possible values can be counted or listedThe number of defective units in a batch of 20A listener rating (on a scale of 1 to 5) in an AccuRating music surveyContinuous random variable: May assume any numerical value in one or more intervals The waiting time for a credit card authorizationThe interest rate charged on a business loanLO5-1: Explain the difference between a discrete random variable and a continuous random variable.5-*5.2 Discrete Probability DistributionsThe 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 assumeNotation: Denote the values of the random variable by x and the value’s associated probability by p(x)LO5-2: Find a discreteprobability distribution and compute its mean and standard deviation.5-*Discrete Probability Distribution PropertiesFor any value x of the random variable, p(x)  0 The probabilities of all the events in the sample space must sum to 1, that isLO5-25-*5.3 The Binomial DistributionThe binomial experiment characteristicsExperiment consists of n identical trialsEach trial results in either “success” or “failure”Probability of success, p, is constant from trial to trialThe probability of failure, q, is 1 – pTrials are independentIf x is the total number of successes in n trials of a binomial experiment, then x is a binomial random variableLO5-3: Use the binomial distribution to compute probabilities.5-*Binomial Distribution ContinuedFor a binomial random variable x, the probability of x successes in n trials is given by the binomial distribution:n! is read as “n factorial” and n! = n × (n-1) × (n-2) × ... × 10! =1Not defined for negative numbers or fractionsLO5-35-*5.4 The Poisson DistributionConsider the number of times an event occurs over an interval of time or space, and assume thatThe probability of occurrence is the same for any intervals of equal lengthThe occurrence in any interval is independent of an occurrence in any non-overlapping intervalIf x = the number of occurrences in a specified interval, then x is a Poisson random variableLO5-4: Use the Poissondistribution to compute probabilities (Optional).5-*5.5 The Hypergometric Distribution (Optional)Population consists of N itemsr of these are successes(N-r) are failuresIf we randomly select n items without replacement, the probability that x of the n items will be successes is given by the hypergeometric probability formulaLO5-5: Use the hypergeometric distribution to compute probabilities (Optional).5-*The Mean and Variance of a Hypergeometric Random VariableLO5-55-*5.6 Joint Distributions and the Covariance (Optional)LO5-6: Compute and understand the covariance between two random variables (Optional).5-*Four Properties of Expected Values and VariancesIf a is a constant and x is a random variable, then μax=aμxIf x1,x2,,xn are random variables, then μ(x1,x2,,xn)= μx1 + μx2 + + μxnIf a is a constant and x is a random variable, then σ2ax=a2σ2xIf x1,x2,,xn are statistically independent random variables, then the covariance is zeroAlso, σ2(x1,x2,,xn)= σ2x1+ σ2x2++ σ2xnLO5-65-*