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McGraw-Hill/Irwin
7
Chapter
Continuous Probability
Distributions
Describing a Continuous Distribution
Uniform Continuous Distribution
Normal Distribution
Standard Normal Distribution
Normal Approximations
Exponential Distribution
Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.
Describing a Continuous Distribution
Continuous Variable – events are intervals and probabilities are areas
underneath smooth curves. A single point has no probability.
PDFs and CDFs
•
•
Probability Density Function
(PDF)
For a continuous
random variable,
the PDF is an
equation that shows
the height of the
curve f(x) at each
possible value of X
over the range of X.
Total area under
curve = 1
7-2
Describing a Continuous Distribution
PDFs and CDFs
•
•
•
Continuous CDF’s:
Denoted F(x)
Shows P(X < x), the
cumulative proportion
of scores
Useful for finding
probabilities
7-3
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.
7-4
Uniform Continuous Distribution
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
7-5
Uniform Continuous Distribution
Characteristics of the Uniform Distribution
7-6
Normal Distribution
Characteristics of the Normal Distribution
•
Normal PDF f(x).
Bell-shaped curve
7-7
Normal Distribution
Characteristics of the Normal Distribution
7-8
Standard Normal Distribution
Characteristics of the Standard Normal
7-9
Normal Approximation to the
Binomial
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 = n
 = n(1- )
7-10
Normal Approximation to the
Binomial
Continuity Correction
•
The table below shows some events and
their cutoff point for the normal
approximation.
7-11
Normal Approximation to the
Poisson
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=
= 
7-12
Exponential Distribution
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.
7-13
Exponential Distribution
Characteristics of the Exponential Distribution
7-14
Exponential Distribution
Characteristics of the Exponential Distribution
Probability of waiting more than x
Probability of waiting less than x
7-15