chapter 5 continuous probability distributions ©

Chapter 5Chapter 5

Continuous Probability Continuous Probability DistributionsDistributions

©

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.

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.

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.

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.

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

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.

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

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.

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

Normal Probability Normal Probability DistributionDistribution

(Figure 5-2)(Figure 5-2)

Mean Median Mode

x

Probability = 0.50Probability = 0.50f(x)

Differences Between Differences Between Normal DistributionsNormal Distributions

(Figure 5-3)(Figure 5-3)

x

x

x

(a)

(b)

(c)

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.

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

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%

Areas Under the Standard Areas Under the Standard Normal CurveNormal Curve

(Table 5-1)(Table 5-1)

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

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

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.

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)(

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

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.

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

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

Exponential ProbabilityExponential Probability

EXPONENTIAL PROBABILITYEXPONENTIAL PROBABILITY

aeaxP 1)(

Key TermsKey Terms

• Continuous Random Variable

• Discrete Random Variable

• Exponential Distribution

• Normal Distribution

• Standard Normal Distribution Standard Normal Table

• Uniform Distribution

• z-Value