7-1. Continuous Distributions Chapter 77 Continuous Variables Continuous Variables Describing a...

Continuous DistributionsContinuous DistributionsContinuous DistributionsContinuous Distributions

Chapter7777

Continuous Variables

Describing a Continuous DistributionDescribing a Continuous Distribution

Uniform Continuous DistributionUniform Continuous Distribution

Normal DistributionNormal Distribution

Standard Normal DistributionStandard Normal Distribution

Normal Approximation to the Binomial (Optional)Normal Approximation to the Binomial (Optional)

Normal Approximation to the Poisson (Optional)Normal Approximation to the Poisson (Optional)

Exponential DistributionExponential DistributionMcGraw-Hill/Irwin © 2008 The McGraw-Hill Companies, Inc. All rights reserved.

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Continuous VariablesContinuous VariablesContinuous VariablesContinuous Variables

• Discrete VariableDiscrete Variable – each value of – each value of XX has its own has its own probability probability PP((XX).).

• Continuous VariableContinuous Variable – events are – events are intervalsintervals and and probabilities are areas underneath smooth probabilities are areas underneath smooth curves. A single point has no probability.curves. A single point has no probability.

Events as IntervalsEvents as Intervals

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Describing a Continuous Describing a Continuous DistributionDistribution


• Probability Density Function (PDF)Probability Density Function (PDF) – – For a continuous For a continuous random variable, random variable, the PDF is an the PDF is an equation that shows equation that shows the height of the the height of the curve curve ff(x) at each (x) at each possible value of possible value of XX over the range of over the range of XX..

PDFs and CDFsPDFs and CDFs

Normal PDF

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Continuous PDF’s:Continuous PDF’s:• Denoted Denoted ff(x)(x)• Must be nonnegativeMust be nonnegative• Total area under Total area under

curve = 1curve = 1• Mean, variance and Mean, variance and

shape depend onshape depend onthe PDF the PDF parametersparameters

• Reveals the shape Reveals the shape of the distributionof the distribution


Normal PDF

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Continuous CDF’s:Continuous CDF’s:• Denoted Denoted FF(x)(x)• Shows Shows PP((X X << xx), the), the

cumulativecumulative proportion proportion of scoresof scores

• Useful for finding Useful for finding probabilitiesprobabilities


Normal CDF

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Continuous probability functions are smooth curves.Continuous probability functions are smooth curves.• Unlike discrete Unlike discrete

distributions, the distributions, the area at any area at any single point = 0.single point = 0.

• The entire area under The entire area under any PDF must be 1.any PDF must be 1.

• Mean is the balanceMean is the balancepoint of the distribution.point of the distribution.

Probabilities as AreasProbabilities as Areas

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Expected Value and VarianceExpected Value and Variance

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Uniform Continuous Uniform Continuous DistributionDistribution


Characteristics of the Uniform DistributionCharacteristics of the Uniform Distribution

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Example: Anesthesia EffectivenessExample: Anesthesia Effectiveness• An oral surgeon injects a painkiller prior to An oral surgeon injects a painkiller prior to

extracting a tooth. Given the varying extracting a tooth. Given the varying characteristics of patients, the dentist views the characteristics of patients, the dentist views the time for anesthesia effectiveness as a uniform time for anesthesia effectiveness as a uniform random variable that takes between 15 minutes random variable that takes between 15 minutes and 30 minutes.and 30 minutes.

• XX is is UU(15, 30)(15, 30)• aa = 15, = 15, bb = 30, find the mean and standard = 30, find the mean and standard

deviation.deviation.

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Example: Anesthesia EffectivenessExample: Anesthesia Effectiveness

= = aa + + bb 2 2 = =

15 + 3015 + 30 2 2 = = 22.5 minutes 22.5 minutes

= = ((bb – – aa))22 12 12 = = 4.33 minutes 4.33 minutes

(30 – 15)(30 – 15)22 12 12==

Find the probability that the anesthetic takes between Find the probability that the anesthetic takes between 20 and 25 minutes.20 and 25 minutes.

PP((cc < < XX < < dd) = () = (dd – – cc)/()/(bb – – aa))

(25 – 20)/(30 – 15)(25 – 20)/(30 – 15)PP(20 < (20 < XX < 25) = < 25) = = 5/15 = 0.3333 or 33.33%= 5/15 = 0.3333 or 33.33%

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Normal DistributionNormal DistributionNormal DistributionNormal Distribution

Characteristics of the Normal DistributionCharacteristics of the Normal Distribution• Normal or Gaussian distribution was named for Normal or Gaussian distribution was named for

German mathematician Karl Gauss (1777 – German mathematician Karl Gauss (1777 – 1855).1855).

• Defined by two parameters, Defined by two parameters, and and • Denoted Denoted NN((, , ))• Domain is –Domain is – < < XX < + < + • Almost all area under the normal curve is Almost all area under the normal curve is

included in the range included in the range – 3 – 3 < < XX < < + 3 + 3

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Characteristics of the Normal DistributionCharacteristics of the Normal Distribution

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What is Normal?What is Normal?A normal random variable should:A normal random variable should:• Be measured on a continuous scale.Be measured on a continuous scale.• Possess clear central tendency.Possess clear central tendency.• Have only one peak (unimodal).Have only one peak (unimodal).• Exhibit tapering tails.Exhibit tapering tails.• Be symmetric about the mean (equal tails).Be symmetric about the mean (equal tails).

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Standard Normal DistributionStandard Normal DistributionStandard Normal DistributionStandard Normal Distribution

Characteristics of the Standard NormalCharacteristics of the Standard Normal• Since for every value of Since for every value of and and , there is a , there is a

different normal distribution, we transform a different normal distribution, we transform a normal random variable to a normal random variable to a standard normal standard normal distributiondistribution with with = 0 and = 0 and = 1 using the = 1 using the formula:formula:

zz = = xx – –

• Denoted Denoted NN(0,1)(0,1)

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Characteristics of the Standard NormalCharacteristics of the Standard Normal

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Finding Areas by using Standardized VariablesFinding Areas by using Standardized Variables• Suppose John took an economics exam and Suppose John took an economics exam and

scored 86 points. The class mean was 75 with a scored 86 points. The class mean was 75 with a standard deviation of 7. What percentile is John standard deviation of 7. What percentile is John in (i.e., find in (i.e., find PP((XX < 86)? < 86)?

zzJohnJohn = = xx – –

== 86 – 7586 – 75 7 7

= 11/7 = 1.57= 11/7 = 1.57

• So John’s score is 1.57 standard deviations about So John’s score is 1.57 standard deviations about the mean. the mean.

7-1. Continuous Distributions Chapter 77 Continuous Variables Continuous Variables Describing a...

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