Chapter 6:Continuous

Distributions

LO1 Solve for probabilities in a continuous uniform distribution.

LO2 Solve for probabilities in a normal distribution using z scores and for the mean, the standard deviation, or a value of x in a normal distribution when given information about the area under the normal curve.

LO3 Solve problems from the discrete binomial distribution using the continuous normal distribution and correcting for continuity.

LO4 Solve for probabilities in an exponential distribution and contrast the exponential distribution to the discrete Poisson distribution.

Learning Objectives

• Continuous distributions are constructed from continuous random variables

• They are generated from experiments or processes that create outcomes that are measured as opposed to counted.

• For continuous distributions probabilities are defined in terms of the likelihood of the event or variable (measurement) occurring in an interval.

• It is always the area under the probability density function in a specified interval

• The probability of a single point, for a continuous distribution is zero. P(x = 5) =0 because it has no area under the curve. This is not the case for the discrete distributions

Continuous Distributions

• The Uniform distribution is a continuous probability density function

• It is defined for the continuous random measures or points occurring with the identical probability, f(x),in a specified closed interval, say [a b]

• The shape of the distribution is rectangular with height = f(x) = 1/(b-a) and width = (b-a)

• The Area under the curve is• The mean and standard deviation of the uniform distribution

are:

The Uniform Distribution

1( )* ( ) ( ) * 1

( )b a Width b a

b a

2

tan12

a bMean

b aS dardDeviation

Uniform Distribution

Determining Probabilities in a Uniform Distribution of Lot Masses

Uniform Distribution ProbabilityDistribution of Lot Masses

Uniform Distribution ProbabilityDistribution of Lot Masses

Uniform DistributionAssembly of Plastic Modules

Uniform DistributionAssembly of Plastic Modules

The height, mean, and standard deviation.

Uniform DistributionAssembly of Plastic Modules

• One of the most widely known and used among all distributions is the normal distribution.

• It applies to many characteristics associated with the elements of a wide range of statistical populations

• In human populations, variables, such as height, weight, length, speed, IQ scores, scholastic achievements, and years of life expectancy, among others tend to be normally distributed.

• Many things in nature such as trees, animals, insects, and others have many characteristics that are normally distributed.

Normal Distribution

Normal Distribution in Business and the Economy

• Many variables in business and industry are also normally distributed.

• Examples are: annual cost of household insurance, the cost of renting warehouse space, managers’ satisfaction rating of support from ownership, amount of fill in soda cans, etc.

• Because of the many applications, the normal distribution is an extremely important distribution.

• Discovery of the normal curve of errors is generally credited to mathematician and astronomer Karl Gauss (1777 – 1855), who recognized that the errors of repeated measurement of objects are often normally distributed.

• Thus the normal distribution is sometimes referred to as the Gaussian distribution or the normal curve of errors.

• In addition, some credit were also given to Pierre-Simon de Laplace (1749 – 1827) and Abraham de Moivre (1667 – 1754) for the discovery of the normal distribution.

Normal Distribution of Errors

The normal distribution exhibits the following characteristics.• It is a continuous distribution.• It is a symmetrical distribution about its mean.• It is asymptotic to the horizontal axis.• It is unimodal.• It is a family of curves.• The area under the curve is 1.

Normal Distribution

Graphic Representation of the Normal Distribution As Bell Shaped

Creating A Family of Normal CurvesKeep either μ or σ constant and varying the other

Standardized Normal Distribution

• In the real world there are a large number of examples of data that can be summarized by the normally distributed.

• Theoretically, there is an infinite number of combinations for and , hence we can generate an infinite family of curves.

• Because of the above , it would be impractical to deal with all of these normal distributions.

• Fortunately, a mechanism was developed by which all normal distributions can be converted into a single distribution called the z distribution.

• This process yields the standardized normal distribution (or curve).

• The conversion formula for any x value of a given normal distribution is given below. It is called the z-score or normal z.

• A z-score gives the number of standard deviations that a value x, is above or below the mean.

• By way of interpretation, z is a standardized measure of the extent to which things differ from the norm or what is expected for a population.

Standardized Normal Distribution

xz

• If x is normally distributed with a mean of and a standard deviation of , then the z-score will also be normally distributed with a mean of 0 and a standard deviation of 1.

• Since we can covert to this standard normal distribution, tables have been generated for this standard normal distribution which will enable us to determine probabilities for normal variables.

• The tables in the text are set up to give the probabilities between z = 0 and some other z value, which is depicted on the next slide.

Standardized Normal Distribution

Probability Density Function of the Normal Distribution

Standardized Normal Distribution

Z Table

Applying the Z Formula

Applying the Z Formula

Applying the Z Formula

Applying the Z Formula

• The normal distribution can be used to approximate binomial probabilities.• Procedure– Convert binomial parameters to normal

parameters.– Does the interval 3 lie between 0 and n? If so,

continue; otherwise, do not use the normal approximation.

– Correct for continuity.– Solve the normal distribution problem.

Normal Approximation of the Binomial Distribution

• Conversion equations

Normal Approximation of Binomial: Parameter Conversion

• Conversion example:

Normal Approximation of Binomial:Interval Check

Graph of the Binomial Problem: n = 60, p = 0.3

x

P(x)

3025201510

0.12

0.10

0.08

0.06

0.04

0.02

0.00

Normal Approximation of Binomial: Correcting for Continuity

Normal Approximation of Binomial: Computations

• Continuous• Family of distributions• Skewed to the right• X varies from 0 to infinity• Apex is always at X = 0• Steadily decreases as X gets larger• Probability function

Exponential Distribution

Graphs of SomeExponential Distributions

Exponential Distribution:Probability Computation

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