chapter 5 continuous probability distributions ©
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Chapter 5Chapter 5
Continuous Probability Continuous Probability DistributionsDistributions
©
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Chapter 5 - Chapter 5 - Chapter Chapter OutcomesOutcomes
After studying the material in this chapter, you should be able to:
• Discuss the important properties of the normal probability distribution.
• Recognize when the normal distribution might apply in a decision-making process.
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Chapter 5 - Chapter 5 - Chapter Chapter OutcomesOutcomes
(continued)(continued)
After studying the material in this chapter, you should be able to:
• Calculate probabilities using the normal distribution table and be able to apply the normal distribution in appropriate business situations.
• Recognize situations in which the uniform and exponential distributions apply.
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Continuous Probability Continuous Probability DistributionsDistributions
A discrete random discrete random variablevariable is a variable that can take on a countable number of possible values along a specified interval.
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Continuous Probability Continuous Probability DistributionsDistributions
A continuous random variablecontinuous random variable is a variable that can take on any of the possible values between two points.
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Examples of Continuous Examples of Continuous Random variablesRandom variables
• Time required to perform a job• Financial ratios• Product weights• Volume of soft drink in a 12-ounce
can• Interest rates• Income levels• Distance between two points
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Continuous Probability Continuous Probability DistributionsDistributions
The probability distribution of a continuous random variable is represented by a probability density functionprobability density function that defines a curve.
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Continuous Probability Continuous Probability DistributionsDistributions
x
f(x)P(x)
Possible Values of x
xPossible Values of x
(a) Discrete Probability Distribution
(b) Probability Density Function
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Normal Probability Normal Probability DistributionDistribution
The Normal DistributionNormal Distribution is a bell-shaped, continuous distribution with the following properties:
1. It is unimodalunimodal.2. It is symmetricalsymmetrical; this means 50% of
the area under the curve lies left of the center and 50% lies right of center.
3. The mean, median, and mode are equal.
4. It is asymptoticasymptotic to the x-axis.5. The amount of variation in the
random variable determines the width of the normal distribution.
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Normal Probability Normal Probability DistributionDistribution
NORMAL DISTRIBUTION DENSITY NORMAL DISTRIBUTION DENSITY FUNCTIONFUNCTION
where:x = Any value of the continuous random
variable = Population standard deviatione = Base of the natural log = 2.7183 = Population mean
22 2/)(
2
1)(
xexf
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Normal Probability Normal Probability DistributionDistribution
(Figure 5-2)(Figure 5-2)
Mean Median Mode
x
Probability = 0.50Probability = 0.50f(x)
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Differences Between Differences Between Normal DistributionsNormal Distributions
(Figure 5-3)(Figure 5-3)
x
x
x
(a)
(b)
(c)
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Standard Normal Standard Normal DistributionDistribution
The standard normal distributionstandard normal distribution is a normal distribution which has a mean = 0.0 and a standard deviation = 1.0. The horizontal axis is scaled in standardized z-values that measure the number of standard deviations a point is from the mean. Values above the mean have positive z-values and those below have negative z-values.
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Standard Normal Standard Normal DistributionDistribution
STANDARDIZED NORMAL Z-VALUESTANDARDIZED NORMAL Z-VALUE
where:x = Any point on the horizontal axis = Standard deviation of the normal
distribution = Population meanz = Scaled value (the number of standard
deviations a point x is from the mean)
x
z
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Areas Under the Standard Areas Under the Standard Normal CurveNormal Curve
(Using Table 5-1)(Using Table 5-1)
X
0.1985
Example:
z = 0.52 (or -0.52)
P(0 < z < .52) = 0.1985 or 19.85%
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Areas Under the Standard Areas Under the Standard Normal CurveNormal Curve
(Table 5-1)(Table 5-1)
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Standard Normal ExampleStandard Normal Example(Figure 5-6)(Figure 5-6)
xx=
zz= -.
Probabilities from the Normal Curve for
Westex
50.010
5045
x
z
0.1915 0.50
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Standard Normal ExampleStandard Normal Example(Figure 5-7)(Figure 5-7)
z
x=7.5
z=-1.25
25.14.0
85.7
x
z
From the normal table: P(-1.25 z 0) = 0.3944
Then, P(x 7.5 hours) = 0.50 - 0.3944 = 0.1056
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Uniform Probability Uniform Probability DistributionDistribution
The uniform distributionuniform distribution is a probability distribution in which the probability of a value occurring between two points, a and b, is the same as the probability between any other two points, c and d, given that the distribution between a and b is equal to the distance between c and d.
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Uniform Probability Uniform Probability DistributionDistribution
CONTINUOUS UNIFORM DISTRIBUTIONCONTINUOUS UNIFORM DISTRIBUTION
where: f(x) = Value of the density function at any x value
a = Lower limit of the interval from a to b
b = Upper limit of the interval from a to b
otherwisexf
bxaifab
xf
0)(
1)(
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Uniform Probability Uniform Probability DistributionsDistributions
(Figure 5-16)(Figure 5-16)
f(x)
2 5a b
.25
.50
f(x)
.25
.50
a b
3 8
33.03
1
25
1)(
xf
for 2 x 5
20.05
1
38
1)(
xf
for 3 x 8
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Exponential Probability Exponential Probability DistributionDistribution
The exponential probability exponential probability distributiondistribution is a continuous distribution that is used to measure the time that elapses between two occurrences of an event.
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Exponential Probability Exponential Probability DistributionDistribution
EXPONENTIAL DISTRIBUTIONEXPONENTIAL DISTRIBUTIONA continuous random variable that is exponentially distributed has the probability density function given by:
where: e = 2.71828. . .
1/ = The mean time between events ( >0)
0,)( xexf x
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Exponential DistributionsExponential Distributions(Figure 5-18)(Figure 5-18)
Values of x
f(x) Lambda = 3.0 (Mean = 0.333)
Lambda = 2.0 (Mean = 0.5)
Lambda = 1.0 (Mean = 1.0)
Lambda = 0.50 (Mean = 020)
x
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Exponential ProbabilityExponential Probability
EXPONENTIAL PROBABILITYEXPONENTIAL PROBABILITY
aeaxP 1)(
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Key TermsKey Terms
• Continuous Random Variable
• Discrete Random Variable
• Exponential Distribution
• Normal Distribution
• Standard Normal Distribution Standard Normal Table
• Uniform Distribution
• z-Value