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4-2 Probability Distributions and
Probability Density Functions
Figure 4-2 Probability determined from the area
under f(x).
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4-2 Probability Distributions and
Probability Density Functions
Definition
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4-2 Probability Distributions and
Probability Density Functions
Figure 4-3 Histogram approximates a probability
density function.
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4-2 Probability Distributions and
Probability Density Functions
Page 7
4-2 Probability Distributions and
Probability Density Functions
Example
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b) What is the probability that a metal cylinder has a diameter
between 49.8 mm and 50.1 mm?
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4-2 Probability Distributions and
Probability Density Functions
Example 4-2
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4-2 Probability Distributions and
Probability Density Functions
Figure 4-5 Probability density function for Example 4-2.
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4-2 Probability Distributions and
Probability Density Functions
Example 4-2 (continued)
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4-3 Cumulative Distribution Functions
Definition
Page 15
4-3 Cumulative Distribution Functions
Example
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4-3 Cumulative Distribution Functions
Example 4-4
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4-3 Cumulative Distribution Functions
Figure 4-7 Cumulative distribution function for Example
4-4.
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4-4 Mean and Variance of a Continuous
Random Variable
Definition
Page 20
4-4 Mean and Variance of a Continuous
Random Variable
Example 4-6
Page 21
4-4 Mean and Variance of a Continuous
Random Variable
Example 4-8
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Example
Suppose that the diameter of a metal cylinder has a pdf
of:
What is the expected value of the cylinder diameter?
elsewhere 0
5.505.49for 5065.12
xf
xxxf
Page 24
4-5 Continuous Uniform Random
Variable
Definition
Page 25
4-5 Continuous Uniform Random
Variable
Figure 4-8 Continuous uniform probability density
function.
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4-5 Continuous Uniform Random
Variable
Mean and Variance
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4-5 Continuous Uniform Random
Variable
Example 4-9
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4-5 Continuous Uniform Random
Variable
Figure 4-9 Probability for Example 4-9.
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4-5 Continuous Uniform Random
Variable
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4-6 Normal Distribution
Definition
Page 31
4-6 Normal Distribution
Figure 4-10 Normal probability density functions for
selected values of the parameters and 2.
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4-6 Normal Distribution
Some useful results concerning the normal distribution
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4-6 Normal Distribution
Definition : Standard Normal
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4-6 Normal Distribution
Example 4-11
Figure 4-13 Standard normal probability density function.
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4-6 Normal Distribution
Standardizing
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4-6 Normal Distribution
Example 4-13
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4-6 Normal Distribution
Figure 4-15 Standardizing a normal random variable.
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4-6 Normal Distribution
To Calculate Probability
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4-6 Normal Distribution
Example 4-14
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4-6 Normal Distribution
Example 4-14 (continued)
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4-6 Normal Distribution
Example 4-14 (continued)
Figure 4-16 Determining the value of x to meet a
specified probability.
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4-7 Normal Approximation to the
Binomial and Poisson Distributions
• Under certain conditions, the normal
distribution can be used to approximate the
binomial distribution and the Poisson
distribution.
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4-7 Normal Approximation to the
Binomial and Poisson Distributions
Figure 4-19 Normal
approximation to the
binomial.
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4-7 Normal Approximation to the
Binomial and Poisson Distributions
Example 4-17
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4-7 Normal Approximation to the
Binomial and Poisson Distributions
Normal Approximation to the Binomial Distribution
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4-7 Normal Approximation to the
Binomial and Poisson Distributions
Example 4-18
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4-7 Normal Approximation to the
Binomial and Poisson Distributions
Figure 4-21 Conditions for approximating hypergeometric
and binomial probabilities.
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4-7 Normal Approximation to the
Binomial and Poisson Distributions
Normal Approximation to the Poisson Distribution
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4-7 Normal Approximation to the
Binomial and Poisson Distributions
Example 4-20
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4-8 Exponential Distribution
Definition
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4-8 Exponential Distribution
More Explanation on the exponential distribution
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4-8 Exponential Distribution
Proof for the pdf of the exponential distribution
Note: The derivation of the distribution of X depends only on the
assumption that the flaws in the wire follow a Poisson Distribution.
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4-8 Exponential Distribution
Mean and Variance
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4-8 Exponential Distribution
Example 4-21
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4-8 Exponential Distribution
Figure 4-23 Probability for the exponential distribution in
Example 4-21.
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4-8 Exponential Distribution
Example 4-21 (continued)
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4-8 Exponential Distribution
Example 4-21 (continued)
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4-8 Exponential Distribution
Example 4-21 (continued)
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4-11 Lognormal Distribution
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4-11 Lognormal Distribution
Figure 4-27 Lognormal probability density functions with = 0
for selected values of 2.
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4-11 Lognormal Distribution
Example 4-26
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4-11 Lognormal Distribution
Example 4-26 (continued)
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4-11 Lognormal Distribution
Example 4-26 (continued)