Chapter 6 covers continuous distributions including the most important Normal distribution. Describe the key features of the normal curve and provide an example of Normal distribution.

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Chapter
6

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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

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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

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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

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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!!!

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6.3 Exponential Probability
Distributions
Normal
Uniform
Exponential
Specific Continuous
Probability Distributions

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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

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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

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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

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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

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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

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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)

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Calculating Exponential Probabilities Using Excel

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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!!!

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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)

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Printed in the United States of America.

0.5
=
< < ¥ - μ) x P( 0.5 = ¥ < < ) x P( μ 1.0 = ¥ < < ¥ - ) x P( 2 2 1 ] (1/2)[ μ)/σ (x e π σ f(x) - - = 0.6449 0.0808 0.7257 ) 48 38 ( = - = £ £ x P x e x f λ λ ) ( - = λ 1 ) ( a e a x P - - = £ λ 1 = = m s