1 1 slide © 2016 cengage learning. all rights reserved. a random variable is a numerical...

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© 2016 Cengage Learning. All Rights Reserved. © 2016 Cengage Learning. All Rights Reserved.

A A random variablerandom variable is a numerical description of the is a numerical description of the outcome of an experiment.outcome of an experiment. A A random variablerandom variable is a numerical description of the is a numerical description of the outcome of an experiment.outcome of an experiment.

Random VariablesRandom Variables

A A discrete random variablediscrete random variable may assume either a may assume either a finite number of values or an infinite sequence offinite number of values or an infinite sequence of values.values.

A A discrete random variablediscrete random variable may assume either a may assume either a finite number of values or an infinite sequence offinite number of values or an infinite sequence of values.values.

A A continuous random variablecontinuous random variable may assume any may assume any numerical value in an interval or collection ofnumerical value in an interval or collection of intervals.intervals.

A A continuous random variablecontinuous random variable may assume any may assume any numerical value in an interval or collection ofnumerical value in an interval or collection of intervals.intervals.

Chapter 5 - Discrete Probability Distributions

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Random Variables…

A A random variablerandom variable is a function or rule that assigns a is a function or rule that assigns a numbernumber to each outcome of an experiment. to each outcome of an experiment. Basically it Basically it is just a symbol that represents the outcome of an is just a symbol that represents the outcome of an experiment.experiment.

X = number of heads when the experiment is flipping a X = number of heads when the experiment is flipping a coin coin 20 times.20 times.

C = the daily change in a stock price. C = the daily change in a stock price.

R = the number of miles per gallon you get on your auto R = the number of miles per gallon you get on your auto during the drive to your family’s home.during the drive to your family’s home.

Y = the amount of sugar in a mountain dew (not diet of Y = the amount of sugar in a mountain dew (not diet of course).course).

V = the speed of an auto registered on a radar detector V = the speed of an auto registered on a radar detector used used on I-83on I-83

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Two Types of Random Variables…

Discrete Random VariableDiscrete Random Variable – usually count data [Number of] – usually count data [Number of]* one that takes on a * one that takes on a countablecountable number of values number of values – this means you – this means you can sit down and list can sit down and list allall possible outcomes without missing any, possible outcomes without missing any, although it might take you an infinite amount of time.although it might take you an infinite amount of time.

X = values on the roll of two dice: X has to be either 2, 3, 4, …, or 12.X = values on the roll of two dice: X has to be either 2, 3, 4, …, or 12.Y = number of customer at Starbucks during the day Y = number of customer at Starbucks during the day Y has to Y has to be 0, 1, 2, 3, 4, 5, 6, 7, 8, ……………”real big number”be 0, 1, 2, 3, 4, 5, 6, 7, 8, ……………”real big number”

Continuous Random VariableContinuous Random Variable – usually measurement data [time, – usually measurement data [time, weight, distance, etc]weight, distance, etc]* one that takes on an uncountable number of values * one that takes on an uncountable number of values – this means – this means you can never list all possible outcomes even if you had an infinite you can never list all possible outcomes even if you had an infinite amount of time.amount of time.

X = time it takes you to walk home from class: X > 0, might be 5.1 X = time it takes you to walk home from class: X > 0, might be 5.1 minutes measured to the nearest tenth but in reality the actual time minutes measured to the nearest tenth but in reality the actual time is 5.10000001…………………. minutes?)is 5.10000001…………………. minutes?)

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Random Variables

QuestionQuestion Random Variable Random Variable xx TypeType

FamilyFamilysizesize

xx = Number of dependents = Number of dependents reported on tax returnreported on tax return

DiscreteDiscrete

Distance fromDistance fromhome to storehome to store

xx = Distance in miles from = Distance in miles from home to the store sitehome to the store site

ContinuousContinuous

Own dogOwn dogor cator cat

xx = 1 if own no pet; = 1 if own no pet; = 2 if own dog(s) only; = 2 if own dog(s) only; = 3 if own cat(s) only; = 3 if own cat(s) only; = 4 if own dog(s) and cat(s)= 4 if own dog(s) and cat(s)

DiscreteDiscrete

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We can describe a discrete probability distribution with a table, We can describe a discrete probability distribution with a table, graph, or formula.graph, or formula. We can describe a discrete probability distribution with a table, We can describe a discrete probability distribution with a table, graph, or formula.graph, or formula.

Discrete Probability Distributions

The probability distribution is defined by a The probability distribution is defined by a probability functionprobability function, , Denoted by Denoted by ff((xx), which provides the probability for each value ), which provides the probability for each value of the random variableof the random variable..

The probability distribution is defined by a The probability distribution is defined by a probability functionprobability function, , Denoted by Denoted by ff((xx), which provides the probability for each value ), which provides the probability for each value of the random variableof the random variable..

The required conditions for a discrete probability function are:The required conditions for a discrete probability function are: The required conditions for a discrete probability function are:The required conditions for a discrete probability function are:

ff((xx) ) >> 0 0ff((xx) ) >> 0 0 ff((xx) = 1) = 1ff((xx) = 1) = 1

The The probability distributionprobability distribution for a random variable for a random variable describes how probabilities are distributed over the describes how probabilities are distributed over the values of the random variable.values of the random variable.

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Discrete Uniform Probability Distribution

The The discrete uniform probability distributiondiscrete uniform probability distribution is the is the simplest example of a discrete probabilitysimplest example of a discrete probability distribution given by a formula.distribution given by a formula.

The The discrete uniform probability distributiondiscrete uniform probability distribution is the is the simplest example of a discrete probabilitysimplest example of a discrete probability distribution given by a formula.distribution given by a formula.

The The discrete uniform probability functiondiscrete uniform probability function is is The The discrete uniform probability functiondiscrete uniform probability function is is

ff((xx) = 1/) = 1/nnff((xx) = 1/) = 1/nnwhere:where:

nn = the number of values the random = the number of values the random variable may assumevariable may assume

Example would be tossing a coin (Head or Tails) and Example would be tossing a coin (Head or Tails) and ff(x) = (1/n) =1/2 or rolling a die where (x) = (1/n) =1/2 or rolling a die where ff(x) = (1/n) = 1/6 (x) = (1/n) = 1/6

the values of the values of thethe

random random variablevariable

are equally are equally likelylikely

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Let Let xx = number of TVs sold at the store in one day, = number of TVs sold at the store in one day,

where where xx can take on 5 values (0, 1, 2, 3, 4) can take on 5 values (0, 1, 2, 3, 4)

Let Let xx = number of TVs sold at the store in one day, = number of TVs sold at the store in one day,

where where xx can take on 5 values (0, 1, 2, 3, 4) can take on 5 values (0, 1, 2, 3, 4)

Example: JSL AppliancesExample: JSL Appliances

Discrete Random Variable with a Finite Number of Values

We can count the TVs sold, and there is a finite upper limit onWe can count the TVs sold, and there is a finite upper limit onthe number that might be sold (the number of TVs in stock).the number that might be sold (the number of TVs in stock).

Discrete Random Variable with an Infinite Sequence of Values

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, . . .

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, . . .

We can count the customers arriving, but there is no finiteWe can count the customers arriving, but there is no finiteupper limit on the number that might arrive.upper limit on the number that might arrive.

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• a a tabular representationtabular representation of the probability of the probability distribution 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 80 80 11 50 50 22 40 40 33 10 10 44 20 20

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 Distributions


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Expected Value

The The expected valueexpected value, or mean, of a random variable, or mean, of a random variable is a measure of its central location.is a measure of its central location.

EE((xx) = ) = = = xfxf((xx))

The The expected valueexpected value, or mean, of a random variable, or mean, of a random variable is a measure of its central location.is a measure of its central location.

EE((xx) = ) = = = xfxf((xx))

The expected value is a weighted average of theThe expected value is a weighted average of the values the random variable may assume. Thevalues the random variable may assume. The weights are the probabilities.weights are the probabilities.

The expected value is a weighted average of theThe expected value is a weighted average of the values the random variable may assume. Thevalues the random variable may assume. The weights are the probabilities.weights are the probabilities.

The expected value does not have to be a value theThe expected value does not have to be a value the random variable can assume.random variable can assume. The expected value does not have to be a value theThe expected value does not have to be a value the random variable can assume.random variable can assume.

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expected number expected number of TVs sold in a of TVs sold in a

dayday

expected number expected number of TVs sold in a of TVs sold in a

dayday

xx ff((xx)) xfxf((xx))

00 .40 .40 .00 .00

11 .25 .25 .25 .25

22 .20 .20 .40 .40

33 .05 .05 .15 .15

44 .10 .10 .40.40

EE((xx) = 1.20) = 1.20

Expected Value


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Variance and Standard Deviation

The The variancevariance summarizes the variability in the summarizes the variability in the values of a random variable.values of a random variable. The The variancevariance summarizes the variability in the summarizes the variability in the values of a random variable.values of a random variable.

The variance is a weighted average of the squaredThe variance is a weighted average of the squared deviations of a random variable from its mean. Thedeviations of a random variable from its mean. The weights are the probabilities.weights are the probabilities.

The variance is a weighted average of the squaredThe variance is a weighted average of the squared deviations of a random variable from its mean. Thedeviations of a random variable from its mean. The weights are the probabilities.weights are the probabilities.

Var(Var(xx) = ) = 22 = = ((xx - - ))22ff((xx))Var(Var(xx) = ) = 22 = = ((xx - - ))22ff((xx))

The The standard deviationstandard deviation, , , is defined as the positive, is defined as the positive square root of the variance.square root of the variance. The The standard deviationstandard deviation, , , is defined as the positive, is defined as the positive square root of the variance.square root of the variance.

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00

11

22

33

44

-1.2-1.2

-0.2-0.2

0.80.8

1.81.8

2.82.8

1.441.44

0.040.04

0.640.64

3.243.24

7.847.84

.40.40

.25.25

.20.20

.05.05

.10.10

.576.576

.010.010

.128.128

.162.162

.784.784

x - x - ((x - x - ))22 ff((xx)) ((xx - - ))22ff((xx))

Variance of daily sales = Variance of daily sales = 22 = 1.660 = 1.660

xx

TVsTVssquaresquare

dd

Standard deviation of daily sales = 1.2884 TVsStandard deviation of daily sales = 1.2884 TVs

Variance Example: JSL AppliancesExample: JSL Appliances

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