Normal Distribution-INSS220

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Chapter6.ppt

Copyright ©2015 Pearson Education, Inc. 6-*

Chapter

6

Copyright ©2015 Pearson Education, Inc. 6-*

Chapter

6

Continuous Probability Distributions

CHAPTER 6 MAP

6.1 Continuous Random Variables

6.2 Normal Probability Distributions

6.3 Exponential Probability Distributions

6.4 Uniform Probability Distributions

Copyright ©2015 Pearson Education, Inc. 6-*

Probability Distributions

Probability Distributions

Discrete

Probability Distributions

Continuous Probability Distributions

Ch. 5

Ch. 6

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6.1 Continuous Random Variables

Continuous random variables are outcomes that take on any numerical value in an interval, as determined by conducting an experiment

  • Usually measured rather than counted
  • Examples of continuous data include time, distance, and weight

The purpose of this chapter is to identify the probability that a specified range of values will occur for continuous random variables, using continuous probability distributions

Copyright ©2015 Pearson Education, Inc. 6-*

Continuous Random Variables

Continuous random variables can take on any value within a specified interval

Because there are an infinite number of possible values, the probability of one specific value occurring is theoretically equal to zero

Probabilities are based on intervals, not individual values

  • Probability is represented by an area under the probability distribution

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Continuous Probability Distributions

The remaining sections in chapter 6 address specific continuous probability distributions

Normal

Uniform

Exponential

Specific Continuous

Probability Distributions

Section 6.2

Section 6.3

Section 6.4

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Continuous Probability Distributions

Continuous probability distributions can have a variety of shapes

Shapes of the three common continuous distributions to be discussed in this chapter:

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Continuous Probability Distributions

The normal probability distribution is useful when the data tend to fall into the center of the distribution and when very high and very low values are fairly rare

The exponential distribution is used to describe data where lower values tend to dominate and higher values don’t occur very often

The uniform distribution describes data where all the values have the same chance of occurring

Normal

Uniform

Exponential

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6.2 Normal Probability Distributions

Normal

Uniform

Exponential

Specific Continuous

Probability Distributions

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Characteristics of the Normal Probability Distribution

  • The distribution is bell-shaped and symmetrical around the mean
  • Because the shape of the

distribution is symmetrical,

the mean and median

are the same value

  • Values near the mean, where

the curve is the tallest, have

a higher likelihood of occurring

than values far from the mean,

where the curve is shorter

Mean

= Median

x

f(x)

μ

σ

Normal Probability Distributions

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Characteristics of the Normal Probability Distribution

  • The total area under the curve is always equal to 1.0

Normal Probability Distributions

f(x)

x

μ

  • Because the distribution is symmetrical around the

mean, the area to the left

of the mean equals 0.5,

as does the area

to the right of the mean

  • The left and right ends of the normal probability distribution extend indefinitely

Copyright ©2015 Pearson Education, Inc. 6-*

Normal Probability Distributions

Changing μ shifts the distribution left or right

Changing σ increases or decreases the spread

x

f(x)

μ

σ2

x

f(x)

μ1

σ1

μ2

σ1 > σ2

  • A distribution’s mean (μ) and standard deviation (σ) completely describe its shape

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Calculating Probabilities for Normal Distributions Using Normal Probability Tables

Any normal distribution (with any mean and standard deviation combination) can be transformed into the standard normal distribution (z)

  • Need to transform x units into z units
  • The resulting z value is called a z-score

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Features of z-scores

  • z-scores are negative for values of x that are less than the distribution mean
  • z-scores are positive for values of x that are more than the distribution mean
  • The z-score at the mean of the distribution equals zero

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The Standard Normal Distribution

When the original random variable, x, follows the normal distribution, z-scores also follow a normal distribution with μ = 0 and σ = 1

This is known as the standard normal distribution

x

f(x)

μ = 0

σ = 1

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Normal Probability Distributions

https://istats.shinyapps.io/NormalDist /

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Normal Probability Distributions

A probability density function is a mathematical description of a probability distribution

  • represents the relative distribution of frequency of a continuous random variable

Formula for the Normal Probability Density Function:

where:

e = 2.71828

π = 3.14159

μ = The mean of the distribution

σ = The standard deviation of the

distribution

x = Any continuous number of

interest

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Calculating Normal Probabilities
Using Excel

Excel’s NORM.DIST function can be used to find normal probabilities

Format for the NORM.DIST function:

= NORM.DIST(x, mean, standard_dev, cumulative)

where:

cumulative is always TRUE for continuous distributions

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Example Using Excel’s NORM.DIST Function

Text tables 3 and 4 only use two decimal places for z-scores

Excel uses more than two decimal places

The difference in reported values is usually small

z

0

x

48

0.60

0.7257

45

If a normal distribution has μ = 45 and σ = 5, what is P(x ≤ 48) ?

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Other Normal Probability Intervals

Example: Probability between two values

Suppose income is normally distributed for a group of workers, with μ = $45,000 and σ = $5,000

Find the probability that a randomly selected worker from this group has an income between $38,000 and $48,000

(Can convert all values to

1000s to simplify)

z

0

Probability = ?

45

38

x

48

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Other Normal Probability Intervals

Example: (continued)

45

38

x

48

45

38

x

48

45

38

x

48

0.7257

0.0808

0.6449

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Using the Normal Distribution to Approximate the Binomial Distribution

The normal distribution can be used as an approximation to the binomial distribution

The normal distribution approximation can be used when the sample size is large enough so that np ≥ 5 and nq ≥ 5

We do NOT discuss it in the class!!!

Copyright ©2015 Pearson Education, Inc. 6-*

6.3 Exponential Probability
Distributions

Normal

Uniform

Exponential

Specific Continuous

Probability Distributions

Copyright ©2015 Pearson Education, Inc. 6-*

Exponential Probability Distributions

The exponential probability distribution is another common continuous distribution

  • Commonly used to measure the time between events of interest
  • Examples:
  • the time between customer arrivals
  • the time between failures in a business process

Copyright ©2015 Pearson Education, Inc. 6-*

Exponential Probability Distributions

Formula for the exponential probability density function:

A discrete random variable that follows the Poisson distribution with a mean equal to λ has a counterpart continuous random variable that follows the exponential distribution with a mean equal to μ = 1/ λ

where:

e = 2.71828

λ = The mean number of occurrences over the interval

x = Any continuous number of interest

Copyright ©2015 Pearson Education, Inc. 6-*

Exponential Probability Distributions

The shape of the exponential distribution depends on the value λ

f(x)

x

λ = 2.0

(mean = 0.5)

λ = 1.0

(mean = 1.0)

λ = 3.0

(mean = .333)

Compared to normal distributions:

  • The exponential distribution is right-skewed, not symmetrical
  • The shape is completely described by only one parameter, λ
  • The values for an exponential random variable cannot be negative

Copyright ©2015 Pearson Education, Inc. 6-*

Exponential Probability Distributions

Formula for the Exponential Cumulative Distribution Function

where:

e = 2.71828

λ = The mean number of occurrences over the interval

a = Any number of interest

Copyright ©2015 Pearson Education, Inc. 6-*

Calculating Exponential Probabilities Using Excel

Excel’s EXPON.DIST function can be used to find exponential probabilities

Format for the EXPON.DIST function:

= EXPON.DIST(x, lambda, cumulative)

where:

cumulative = TRUE

Copyright ©2015 Pearson Education, Inc. 6-*

Exponential Probability Distributions

Formula for the standard deviation of the Exponential Distribution:

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Calculating Exponential Probabilities

Example: The mean time between arrivals is 2 minutes

What is the probability that the next arrival is within the next 3 minutes?

  • Time between arrivals is exponentially distributed with mean time between arrivals of 2 minutes (30 per 60 minutes, on average)

Copyright ©2015 Pearson Education, Inc. 6-*

Calculating Exponential Probabilities Using Excel

Copyright ©2015 Pearson Education, Inc. 6-*

6.4 Uniform Probability Distributions

Normal

Uniform

Exponential

Specific Continuous

Probability Distributions

f(x)

x

55

0.20

0.01

155

70 90

We do NOT discuss it in the class!!!

Copyright ©2015 Pearson Education, Inc. 6-*

Normal Distribution

Exponential Distribution

Probability on left = Value from Excel

Probability on Right = 1 – Value from Excel

Probability in between x1 and x2 = Value from Excel for x2 –Value from Excel for x1

Mean µ
Standard Deviation σ
Probability on left NORM.DIST(x, mean, standard_dev, TRUE)
Mean 1/λ
Standard Deviation 1/λ
Probability on left EXPON.DIST(x, lambda, TRUE)

Copyright ©2015 Pearson Education, Inc. 6-*

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Printed in the United States of America.

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