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Applied Statistics in Business &
Economics, 4th edition
David P. Doane and Lori E. Seward
Prepared by Lloyd R. Jaisingh
McGraw-Hill/Irwin
Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved.
Chapter 7
Continuous Probability Distributions
Chapter Contents
7.1
7.2
7.3
7.4
7.5
7.6
7.7
Describing a Continuous Distribution
Uniform Continuous Distribution
Normal Distribution
Standard Normal Distribution
Normal Approximations
Exponential Distribution
Triangular Distribution (Optional)
7-2
Chapter 7
Continuous Probability Distributions
Chapter Learning Objectives (LO’s)
LO7-1: Define a continuous random variable.
LO7-2: Calculate uniform probabilities.
LO7-3: Know the form and parameters of the normal distribution.
LO7-4: Find the normal probability for given z or x using tables or Excel.
LO7-5: Solve for z or x for a given normal probability using tables or Excel.
7-3
Chapter 7
Continuous Probability Distributions
Chapter Learning Objectives (LO’s)
LO6: Use the normal approximation to a binomial or a Poisson
distribution.
LO7: Find the exponential probability for a given x.
LO8: Solve for x for given exponential probability.
LO9: Use the triangular distribution for “what-if” analysis (optional).
7-4
7.1 Describing a Continuous Distribution
Chapter 7
LO7-1
LO7-1: Define a continuous random variable.
Events as Intervals
•
•
Discrete Variable – each value of X has its own probability P(X).
Continuous Variable – events are intervals and probabilities are
areas under continuous curves. A single point has no probability.
7-5
7.1 Describing a Continuous Distribution
PDF – Probability Density Function
Chapter 7
LO7-1
Continuous PDF’s:
•
Denoted f(x)
•
Must be nonnegative
•
Total area under
curve = 1
•
Mean, variance and
shape depend on
the PDF parameters
•
Reveals the shape
of the distribution
7-6
7.1 Describing a Continuous Distribution
CDF – Cumulative Distribution Function
Chapter 7
LO7-1
Continuous CDF’s:
•
•
•
Denoted F(x)
Shows P(X ≤ x), the
cumulative proportion
of scores
Useful for finding
probabilities
7-7
7.1 Describing a Continuous Distribution
Probabilities as Areas
Chapter 7
LO7-1
Continuous probability functions:
•
•
•
Unlike discrete
distributions, the
probability at any
single point = 0.
The entire area under
any PDF, by definition,
is set to 1.
Mean is the balance
point of the distribution.
7-8
7.1 Describing a Continuous Distribution
Expected Value and Variance
Chapter 7
LO7-1
The mean and variance of a continuous random variable are analogous to
E(X) and Var(X ) for a discrete random variable, Here the integral sign
replaces the summation sign. Calculus is required to compute the integrals.
7-9
Chapter 7
7.2 Uniform Continuous Distribution
LO7-2
LO7-2: Calculate uniform probabilities.
Characteristics of the Uniform
Distribution
If X is a random variable that is
uniformly distributed between
a and b, its PDF has
constant height.
•
•
Denoted U(a, b)
Area =
base x height =
(b-a) x 1/(b-a) = 1
7-10
7.2 Uniform Continuous Distribution
Characteristics of the Uniform Distribution
Chapter 7
LO7-2
7-11
7.2 Uniform Continuous Distribution
Example: Anesthesia Effectiveness
•
•
•
•
Chapter 7
LO7-2
An oral surgeon injects a painkiller prior to extracting a tooth. Given the
varying characteristics of patients, the dentist views the time for
anesthesia effectiveness as a uniform random variable that takes
between 15 minutes and 30 minutes.
X is U(15, 30)
a = 15, b = 30, find the mean and standard deviation.
Find the probability that the effectiveness anesthetic takes between
20 and 25 minutes.
7-12
7.2 Uniform Continuous Distribution
Example: Anesthesia Effectiveness
Chapter 7
LO7-2
P(20 < X < 25) = (25 – 20)/(30 – 15) = 5/15 = 0.3333 = 33.33%
7-13
7.3 Normal Distribution
Chapter 7
LO7-3
LO7-3: Know the form and parameters of the normal distribution.
Characteristics of the Normal Distribution
•
•
•
•
•
•
Normal or Gaussian (or bell shaped) distribution was named for German
mathematician Karl Gauss (1777 – 1855).
Defined by two parameters, µ and .
Denoted N(µ, ).
Domain is –  < X < +  (continuous scale).
Almost all (99.7%) of the area under the normal curve is included in the
range µ – 3 < X < µ + 3.
Symmetric and unimodal about the mean.
7-14
7.3 Normal Distribution
Chapter 7
LO7-3
Characteristics of the Normal Distribution
7-15
Chapter 7
LO7-3
7.3 Normal Distribution
Characteristics of the Normal Distribution
•
Normal PDF f(x) reaches a maximum at µ and has points of inflection at
µ ±
Bell-shaped curve
NOTE: All normal
distributions
have the same
shape but differ
in the axis scales.
7-16
7.3 Normal Distribution
Chapter 7
LO7-3
Characteristics of the Normal Distribution
•
Normal CDF
7-17
7.4 Standard Normal Distribution
Characteristics of the Standard Normal Distribution
•
Chapter 7
LO7-3
Since for every value of µ and , there is a different normal distribution, we transform a normal
random variable to a standard normal distribution with µ = 0 and  = 1 using the formula.
7-18
7.4 Standard Normal Distribution
Characteristics of the Standard Normal
•
Standard normal PDF f(x) reaches a maximum at z = 0 and has
points of inflection at +1.
•
Shape is unaffected by
the transformation.
It is still a bell-shaped
curve.
Chapter 7
LO7-3
Figure 7.11
7-19
7.4 Standard Normal Distribution
Characteristics of the Standard Normal
•
Standard normal CDF
•
•
•
•
Chapter 7
LO7-3
A common scale
from -3 to +3 is used.
Entire area under the
curve is unity.
The probability of an
event P(z1 < Z < z2)
is a definite integral
of f(z).
However, standard
normal tables or
Excel functions can
be used to find the
desired probabilities.
7-20
7.4 Standard Normal Distribution
Normal Areas from Appendix C-1
•
•
Chapter 7
LO7-3
Appendix C-1 allows you to find the area under the curve
from 0 to z.
For example, find P(0 < Z < 1.96):
7-21
7.4 Standard Normal Distribution
Normal Areas from Appendix C-1
•
•
Now find P(-1.96 < Z < 1.96).
Due to symmetry, P(-1.96 < Z) is the same as P(Z < 1.96).
•
So, P(-1.96 < Z < 1.96) = .4750 + .4750 = .9500 or 95% of the
area under the curve.
Chapter 7
LO7-3
7-22
7.4 Standard Normal Distribution
Basis for the Empirical Rule
•
•
•
Chapter 7
LO7-3
Approximately 68% of the area under the curve is between + 1
Approximately 95% of the area under the curve is between + 2
Approximately 99.7% of the area under the curve is between + 3
7-23
7.4 Standard Normal Distribution
LO7-4: Find the normal probability for given z or x using tables or Excel.
Chapter 7
LO7-4
Normal Areas from Appendix C-2
•
Appendix C-2 allows you to find the area under the curve from the left of
z (similar to Excel).
•
For example,
P(Z < 1.96)
P(Z < -1.96)
P(-1.96 < Z < 1.96)
7-24
7.4 Standard Normal Distribution
Normal Areas from Appendices C-1 or C-2
•
•
Chapter 7
LO7-4
Appendices C-1 and C-2 yield identical results.
Use whichever table is easiest.
Finding z for a Given Area
•
•
•
Appendices C-1 and C-2 can be used to find the
z-value corresponding to a given probability.
For example, what z-value defines the top 1% of a normal
distribution?
This implies that 49% of the area lies between 0 and z which
gives z = 2.33 by looking for an area of 0.4900 in Appendix C-1.
7-25
7.4 Standard Normal Distribution
Finding Areas by using Standardized Variables
•
Chapter 7
LO7-4
Suppose John took an economics exam and scored 86 points. The class
mean was 75 with a standard deviation of 7. What percentile is John in?
That is, what is P(X < 86) where X represents the exam scores?
•
So John’s score is 1.57 standard deviations about the mean.
•
P(X < 86) = P(Z < 1.57) = .9418 (from Appendix C-2)
•
So, John is approximately in the 94th percentile.
7-26
•
7.4 Standard Normal Distribution
Finding Areas by using Standardized Variables
Chapter 7
LO7-4
NOTE: You can use Excel, Minitab, TI83/84 etc. to compute these
probabilities directly.
7-27
7.4 Standard Normal Distribution
LO7-5: Solve for z or x for a normal probability using tables or Excel.
•
Chapter 7
LO7-5
Inverse Normal
• How can we find the various normal percentiles (5th, 10th, 25th, 75th,
90th, 95th, etc.) known as the inverse normal? That is, how can we
find X for a given area? We simply turn the standardizing
transformation around:
Solving for x in z = (x − μ)/ gives x = μ + zσ
7-28
•
7.4 Standard Normal Distribution
Chapter 7
LO7-5
Inverse Normal
• For example, suppose that John’s economics professor has decided
that any student who scores below the 10th percentile must retake the
exam.
• The exam scores are normal with μ = 75 and σ = 7.
• What is the score that would require a student to retake the exam?
• We need to find the value of x that satisfies P(X < x) = .10.
• The z-score for with the 10th percentile is z = −1.28.
7-29
•
7.4 Standard Normal Distribution
Chapter 7
LO7-5
Inverse Normal
• The steps to solve the problem are:
• Use Appendix C or Excel to find z = −1.28 to satisfy P(Z < −1.28) = .10.
• Substitute the given information into z = (x − μ)/σ to get
−1.28 = (x − 75)/7
• Solve for x to get x = 75 − (1.28)(7) = 66.03 (or 66 after rounding)
• Students who score below 66 points on the economics exam will be
required to retake the exam.
7-30
•
7.4 Standard Normal Distribution
Inverse Normal
Chapter 7
LO7-5
7-31
7.5 Normal Approximations
LO7-6: Use the normal approximation to a binomial or a Poisson.
Chapter 7
LO7-6
Normal Approximation to the Binomial
•
•
•
Binomial probabilities are difficult to calculate when n is large.
Use a normal approximation to the binomial distribution.
As n becomes large, the binomial bars become smaller and continuity is
approached.
7-32
7.5 Normal Approximations
Normal Approximation to the Binomial
•
•
Chapter 7
LO7-6
Rule of thumb: when n ≥ 10 and n(1- ) ≥ 10, then it is appropriate to use
the normal approximation to the binomial distribution.
In this case, the mean and standard deviation for the binomial distribution
will be equal to the normal µ and , respectively.
Example Coin Flips
•
If we were to flip a coin n = 32 times and  = .50, are the
requirements for a normal approximation to the binomial distribution
met?
7-33
Example Coin Flips
•
•
•
•
•
•
Chapter 7
7.5 Normal Approximations
LO7-6
n = 32 x .50 = 16
n(1- ) = 32 x (1 - .50) = 16
So, a normal approximation can be used.
When translating a discrete scale into a continuous scale,
care must be taken about individual points.
For example, find the probability of more than 17 heads in
32 flips of a fair coin.
This can be written as P(X  18).
However, “more than 17” actually falls between 17 and 18
on a discrete scale.
7-34
7.5 Normal Approximations
Example Coin Flips
•
•
•
Chapter 7
LO7-6
Since the cutoff point for “more than 17” is halfway between 17 and 18, we
add 0.5 to the lower limit and find P(X > 17.5).
This addition to X is called the Continuity Correction.
At this point, the problem can be completed as any normal distribution
problem.
7-35
7.5 Normal Approximations
Example Coin Flips
Chapter 7
LO7-6
P(X > 17) = P(X ≥ 18)  P(X ≥ 17.5)
= P(Z > 0.53) = 0.2981
7-36
7.5 Normal Approximations
Normal Approximation to the Poisson
•
•
Chapter 7
LO7-6
The normal approximation to the Poisson distribution works best
when  is large (e.g., when  exceeds the values in Appendix B).
Set the normal µ and  equal to the mean and standard deviation
for the Poisson distribution.
Example Utility Bills
•
•
•
On Wednesday between 10A.M. and noon customer billing
inquiries arrive at a mean rate of 42 inquiries per hour at
Consumers Energy. What is the probability of receiving more
than 50 calls in an hour?
 = 42 which is too big to use the Poisson table.
Use the normal approximation with  = 42 and  = 6.48074
7-37
7.5 Normal Approximations
Example Utility Bills
•
•
Chapter 7
LO7-6
To find P(X > 50) calls, use the continuity-corrected cutoff point halfway
between 50 and 51 (i.e., X = 50.5).
At this point, the problem can be completed as any normal distribution
problem.
7-38
Chapter 7
LO7-7
7.6 Exponential Distribution
LO7-7: Find the exponential probability for a given x.
Characteristics of the Exponential Distribution
•
•
If events per unit of time follow a Poisson distribution, the time until the
next event follows the Exponential distribution.
The time until the next event is a continuous variable.
NOTE: Here
we will find
probabilities
> x or ≤ x.
7-39
7.6 Exponential Distribution
Characteristics of the Exponential Distribution
Probability of waiting less than or
equal to x
Chapter 7
LO7-7
Probability of waiting more than x
7-40
7.6 Exponential Distribution
Example Customer Waiting Time
•
•
•
•
Chapter 7
LO7-7
Between 2P.M. and 4P.M. on Wednesday, patient insurance
inquiries arrive at Blue Choice insurance at a mean rate of 2.2 calls
per minute.
What is the probability of waiting more than 30 seconds (i.e., 0.50
minutes) for the next call?
Set  = 2.2 events/min and x = 0.50 min
P(X > 0.50) = e–x = e–(2.2)(0.5) = .3329
or 33.29% chance of waiting more than 30 seconds for the next
call.
7-41
7.6 Exponential Distribution
Chapter 7
LO7-7
Example Customer Waiting Time
P(X > 0.50)
P(X ≤ 0.50)
7-42
7.6 Exponential Distribution
LO7-8: Solve for x for given exponential probability.
Chapter 7
LO7-8
Inverse Exponential
•
•
If the mean arrival rate is 2.2 calls per minute, we want the 90th
percentile for waiting time (the top 10% of waiting time).
Find the x-value
that defines the
upper 10%.
7-43
7.6 Exponential Distribution
Inverse Exponential
Chapter 7
LO7-8
7-44
7.6 Exponential Distribution
Chapter 7
LO7-8
Mean Time Between Events
7-45
7.7 Triangular Distribution
LO7-9: Use the triangular distribution for “what-if” analysis (optional).
Chapter 7
LO7-9
Characteristics of the Triangular Distribution
7-46
7.7 Triangular Distribution
Characteristics of the Triangular Distribution
Chapter 7
LO7-9
• The triangular distribution is a way of thinking about variation that
corresponds rather well to what-if analysis in business.
• It is not surprising that business analysts are attracted to the triangular
model.
• Its finite range and simple form are more understandable than a normal
distribution.
7-47
7.7 Triangular Distribution
Characteristics of the Triangular Distribution
Chapter 7
LO7-9
• It is more versatile than a normal, because it can be skewed in either
direction.
• Yet it has some of the nice properties of a normal, such as a distinct mode.
• The triangular model is especially handy for what-if analysis when the
business case depends on predicting a stochastic variable (e.g., the price
of a raw material, an interest rate, a sales volume).
• If the analyst can anticipate the range (a to c) and most likely value (b), it will
be possible to calculate probabilities of various outcomes.
• Many times, such distributions will be skewed, so a normal wouldn’t
be much help.
7-48