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A A random variablerandom variable is a numerical description of theis a numerical description of theoutcome of an experiment.outcome of an experiment.
Random VariablesRandom Variables
A A discrete random variablediscrete random variable may assume either amay assume either afinite number of values or an infinite sequence offinite number of values or an infinite sequence ofvalues.values.
A A continuous random variablecontinuous random variable may assume anymay assume anynumerical value in an interval or collection ofnumerical value in an interval or collection ofintervals.intervals.
Let Let xx = number of customers arriving in one day,= number of customers arriving in one day,
where where xx can take on the values 0, 1, 2, . . .can take on the values 0, 1, 2, . . .
Example: JSL AppliancesExample: JSL Appliances
■■ Discrete random variable with an Discrete random variable with an infiniteinfinite sequence sequence of valuesof values
We can count the customers arriving, but there is noWe can count the customers arriving, but there is nofinite upper limit on the number that might arrive.finite upper limit on the number that might arrive.
The The probability distributionprobability distribution for a random variablefor a random variabledescribes how probabilities are distributed overdescribes how probabilities are distributed overthe values of the random variable.the values of the random variable.
We can describe a discrete probability distributionWe can describe a discrete probability distributionwith a table, graph, or equation.with a table, graph, or equation.
Discrete Probability DistributionsDiscrete Probability Distributions
The probability distribution is defined by aThe probability distribution is defined by aprobability functionprobability function, denoted by , denoted by ff((xx), which provides), which providesthe probability for each value of the random variable.the probability for each value of the random variable.
The required conditions for a discrete probabilityThe required conditions for a discrete probabilityfunction are:function are:
Discrete Probability DistributionsDiscrete Probability Distributions
■■ a tabular representation of the probabilitya tabular representation of the probabilitydistribution for TV sales was developed.distribution for TV sales was developed.
■■ Using past data on TV sales, …Using past data on TV sales, …
NumberNumber
Units SoldUnits Sold of Daysof Days
00 8080
11 5050
22 4040
33 1010
44 2020
200200
xx ff((xx))
00 .40.40
11 .25.25
22 .20.20
33 .05.05
44 .10.10
1.001.00
80/20080/200
Discrete Probability DistributionsDiscrete Probability Distributions
0 1 2 3 40 1 2 3 4Values of Random Variable Values of Random Variable xx (TV sales)(TV sales)Values of Random Variable Values of Random Variable xx (TV sales)(TV sales)
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Discrete Probability DistributionsDiscrete Probability Distributions
■■ Graphical Representation of Probability DistributionGraphical Representation of Probability Distribution
Discrete Uniform Probability DistributionDiscrete Uniform Probability Distribution
The The discrete uniform probability distributiondiscrete uniform probability distribution is theis thesimplest example of a discrete probabilitysimplest example of a discrete probabilitydistribution given by a formula.distribution given by a formula.
The The discrete uniform probability functiondiscrete uniform probability function isis
ff((xx) = 1/) = 1/nn
where:where:nn = the number of values the random= the number of values the random
variable may assumevariable may assume
the values of thethe values of therandom variablerandom variableare equally likelyare equally likely
Expected Value and VarianceExpected Value and Variance
The The expected valueexpected value, or mean, of a random variable, or mean, of a random variableis a measure of its central location.is a measure of its central location.
The The variancevariance summarizes the variability in thesummarizes the variability in thevalues of a random variable.values of a random variable.
The The standard deviationstandard deviation, , σσ, is defined as the positive, is defined as the positivesquare root of the variance.square root of the variance.
■■ Four Properties of a Binomial ExperimentFour Properties of a Binomial Experiment
3. The probability of a success, denoted by 3. The probability of a success, denoted by pp, does, doesnot change from trial to trial.not change from trial to trial.
4. The trials are independent.4. The trials are independent.
2. Two outcomes, 2. Two outcomes, successsuccess and and failurefailure, are possible, are possibleon each trial.on each trial.
1. The experiment consists of a sequence of 1. The experiment consists of a sequence of nnidentical trials.identical trials.
■■ Binomial Probability FunctionBinomial Probability Function
Probability of a particularProbability of a particularsequence of trial outcomessequence of trial outcomeswith x successes in with x successes in nn trialstrials
Number of experimentalNumber of experimentaloutcomes providing exactlyoutcomes providing exactly
Evans is concerned about a low retention rate for Evans is concerned about a low retention rate for employees. In recent years, management has seen a employees. In recent years, management has seen a turnover of 10% of the hourly employees annually. turnover of 10% of the hourly employees annually. Thus, for any hourly employee chosen at random, Thus, for any hourly employee chosen at random, management estimates a probability of 0.1 that the management estimates a probability of 0.1 that the person will not be with the company next year.person will not be with the company next year.
■■ Using the Binomial Probability FunctionUsing the Binomial Probability Function
Choosing 3 hourly employees at random, what is Choosing 3 hourly employees at random, what is the probability that 1 of them will leave the company the probability that 1 of them will leave the company this year?this year?
A Poisson distributed random variable is oftenA Poisson distributed random variable is oftenuseful in estimating the number of occurrencesuseful in estimating the number of occurrencesover a over a specified interval of time or spacespecified interval of time or space
It is a discrete random variable that may assumeIt is a discrete random variable that may assumean an infinite sequence of valuesinfinite sequence of values (x = 0, 1, 2, . . . ).(x = 0, 1, 2, . . . ).
■■ Two Properties of a Poisson ExperimentTwo Properties of a Poisson Experiment
2.2. The occurrence or nonoccurrence in anyThe occurrence or nonoccurrence in anyinterval is independent of the occurrence orinterval is independent of the occurrence ornonoccurrence in any other interval.nonoccurrence in any other interval.
1.1. The probability of an occurrence is the sameThe probability of an occurrence is the samefor any two intervals of equal length.for any two intervals of equal length.
A property of the Poisson distribution is thatA property of the Poisson distribution is thatthe mean and variance are equal.the mean and variance are equal.
Hypergeometric DistributionHypergeometric Distribution
The The hypergeometric distributionhypergeometric distribution is closely relatedis closely relatedto the binomial distribution. to the binomial distribution.
However, for the hypergeometric distribution:However, for the hypergeometric distribution:
the trials are not independent, andthe trials are not independent, and
the probability of success changes from trialthe probability of success changes from trialto trial. to trial.
Hypergeometric DistributionHypergeometric Distribution
■■ Example: NevereadyExample: Neveready
Bob Neveready has removed twoBob Neveready has removed two
dead batteries from a flashlight anddead batteries from a flashlight and
inadvertently mingled them withinadvertently mingled them with
the two good batteries he intendedthe two good batteries he intended
as replacements. as replacements. The four batteries look identical.The four batteries look identical.
Bob now randomly selects two of the four Bob now randomly selects two of the four batteries. What is the probability he selects the two batteries. What is the probability he selects the two good batteries?good batteries?
Hypergeometric DistributionHypergeometric Distribution
■■ Using the Hypergeometric FunctionUsing the Hypergeometric Function
2 2 2! 2!
2 0 2!0! 0!2! 1( ) .167
4 4! 6
2 2!2!
r N r
x n xf x
N
n
− − = = = = =
where:where:xx = 2 = number of = 2 = number of goodgood batteries selectedbatteries selectednn = 2 = number of batteries selected= 2 = number of batteries selectedNN = 4 = number of batteries in total= 4 = number of batteries in totalrr = 2 = number of = 2 = number of goodgood batteries in totalbatteries in total
Hypergeometric DistributionHypergeometric Distribution
Consider a hypergeometric distribution with Consider a hypergeometric distribution with nn trialstrialsand let and let pp = (= (rr//nn) denote the probability of a success) denote the probability of a successon the first trial.on the first trial.
If the population size is large, the term (If the population size is large, the term (NN –– nn)/()/(NN –– 1)1)
approaches 1.approaches 1.
The expected value and variance can be writtenhe expected value and variance can be written
EE((xx) = ) = npnp and and VarVar((xx) = ) = npnp(1 (1 –– pp).).
Note that these are the expressions for the expectedhe expected
value and variance of a binomial distribution.value and variance of a binomial distribution.
Hypergeometric DistributionHypergeometric Distribution
When the population size is large, a hypergeometricWhen the population size is large, a hypergeometricdistribution can be approximated by a binomialdistribution can be approximated by a binomialdistribution with distribution with nn trials and a probability of successtrials and a probability of successpp = (= (rr//NN). ).