Transcript Chapter 7 - McGraw Hill Higher Education

A PowerPoint Presentation Package to Accompany
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
LO7-1:
LO7-2:
LO7-3:
LO7-4:
LO7-5:
LO7-6:
LO7-7:
LO7-8:
LO7-9:
Define a continuous random variable.
Calculate uniform probabilities.
Know the form and parameters of the normal distribution.
Find the normal probability for given x using tables or Excel.
Solve for z or x for a given normal probability using tables or Excel.
Use the normal approximation to a binomial or a Poisson.
Find the exponential probability for a given x.
Solve for x for given exponential probability.
Use the triangular distribution for “what-if” analysis (optional).
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Chapter 7
LO7-1
7.1 Describing a Continuous Distribution
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 underneath
smooth curves. A single point has no probability.
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Chapter 7
LO7-1
7.1 Describing a Continuous Distribution
PDF – Probability Density Function
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.
Normal PDF
CDF – Cumulative Distribution Function
Continuous CDF’s:
•
Denoted F(x).
•
Shows P(X ≤ x), the cumulative
proportion of scores.
•
Useful for finding probabilities.
Normal CDF
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Chapter 7
LO7-1
7.1 Describing a Continuous Distribution
Probabilities as Areas
Continuous probability functions are smooth curves.
•
Unlike discrete distributions, the area at any single point = 0.
•
The entire area under any PDF must be 1.
•
Mean is the balance point of the distribution.
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.
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Chapter 7
LO7-2
7.2 Uniform Continuous Distribution
LO7-2: Calculate uniform probabilities.
Characteristics of the Uniform Distribution
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Chapter 7
LO7-3
7.3 Normal Distribution
LO7-3: Know the form and parameters of the normal distribution.
Characteristics of the Normal Distribution
1. Normal or Gaussian
(or bell shaped) distribution
was named for German
mathematician Karl Gauss
(1777 – 1855).
2. Defined by two parameters,
µ and . Denoted by N(µ,
).
3. Domain is –  < X < + 
(continuous scale)
4. Almost all area under the
normal curve is included in
the range µ – 3 < X < µ +
3.
5. Symmetric and unimodal
7-8
about the mean.
Chapter 7
LO7-3
7.4 Standard Normal Distribution
Characteristics of the Standard Normal Distribution
•
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.
7-9
7.4 Standard Normal Distribution
LO7-4: Find the normal probability for given z or x using tables or Excel.
Normal Areas from Appendices C-1 or C-2
•
•
Appendices C-1 and C-2 will yield identical results.
Use whichever table is easiest.
Finding z for a Given Area
•
Appendices C-1 and C-2 be used to find the z-value corresponding to a given
probability.
LO7-5: Solve for z or x for a normal probability using tables or Excel.
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:
x = μ + zσ solving for x in z = (x − μ)/.
Figure 7.15
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Chapter 7
LO7-4
•
Chapter 7
LO7-5
7.4 Standard Normal Distribution
Inverse Normal
Figure 7.18
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Chapter 7
LO7-6
7.5 Normal Approximations
LO7-6: Use the normal approximation to a binomial or a Poisson.
When is Approximation Needed?
•
•
Rule of thumb: when n > 10 and n(1- ) > 10, then it is appropriate to use the
normal approximation to the binomial.
In this case, the binomial mean and standard deviation will be equal to the normal µ
and , respectively.
μ  nπ
σ  nπ (1  π)
When is Approximation Needed?
•
•
The normal approximation to the Poisson works best when  is large
(e.g., when  exceeds the values in Appendix B).
Set the normal µ and  equal to the Poisson mean and standard
deviation.
μλ
σ
λ
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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 waiting time until
the next event follows the Exponential distribution.
Waiting time until the next event is a continuous variable.
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Chapter 7
LO7-7
7.6 Exponential Distribution
Characteristics of the Exponential Distribution
Probability of waiting more than x
Probability of waiting less than x
LO7-8: Solve for x for given exponential probability.
Inverse Exponential
•
•
For example, 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%.
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Chapter 7
LO7-9
7.7 Triangular Distribution
LO7-9: Use the triangular distribution for “what-if” analysis (optional).
Characteristics of the Triangular Distribution
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Chapter 7
LO7-9
7.7 Triangular Distribution
Characteristics of the Triangular Distribution
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.
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.
7-16