#### Transcript Ch. 20

```Lecture Slides
Chapter 20
Statistical Considerations
Chapter Outline
Shigley’s Mechanical Engineering Design
Random Variables
A random experiment is an experiment in which a number of
specimens are selected at random from a larger batch.
 A random variable is a variable quantity whose value depends on
the outcome of a random experiment.
 A random variable is also called a stochastic variable.
 A sample space contains all the possible outcomes.
 An example sample space for a pair of dice,

Fig. 20–1
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Probability Distribution
The probability of a random variable x taking on a specific value is
called p = f (x).
 p is the number of times a specific x occurs divided by the total
number of possible outcomes.
 A list of all possible values along with the corresponding
probabilities is called a probability distribution.
 Define a random variable x as the sum of numbers obtained from
tossing a pair of dice. The probability distribution for x,

Table 20–1
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Frequency Distribution
Plotting the probability distribution makes it clear that the
probability is a function of x.
 The probability function p = f (x) is often called the frequency
function, or the probability density function (PDF).

Fig. 20–2
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Cumulative Probability Distribution
The probability that x is less than or equal to a certain value of xi is
found by summing the probability of all x’s up to and including xi.
 This results in a cumulative probability distribution.

Table 20–2

The function F(x) is a cumulative density function (CDF) of x.
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Cumulative Frequency Distribution

A plot of the cumulative density function is called a cumulative
frequency distribution.
Fig. 20–3
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Continuous Random Variables

A discrete random variable x can only have discrete values.
A continuous random variable x can have any value in a specified
interval.
For continuous random variables, the distribution plots are
continuous curves.
For a continuous probability density function F(x),

When x goes to ∞,

Differentiation of Eq. (20-2) gives



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Arithmetic Mean, Variance, and Standard Deviation

Sample mean of N elements

Sample variance

Sample standard deviation

Alternate form
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Population Mean and Standard Deviation
If the entire population is considered,x and sx are replaced with mx
and ˆ x respectively.
 The N – 1 in the denominator is replaced by N.

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Discrete Frequency Histogram

A discrete frequency histogram gives the number of occurrences,
or class frequency fi, within a given range.
Fig. 20–4
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Mean and Standard Deviation with Class Frequency

When the data are grouped by class frequency fi, the mean and
standard deviation can be expressed as

The cumulative density function that gives the probability of an
occurrence at class mark xi or less is
Where wi is the class width at xi.
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Notation
Boldface characters indicate random variables that can be
characterized by a mean and a standard deviation.
 The terms stochastic variable or variate are used to mean a
random variable.
 A deterministic quantity is something that has a single specified
value.
 A coefficient of variation is defined by


A variate x can be expressed in the following two ways:
where X represents a variate probability distribution function.
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Example 20–1
Table 20–3
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Example 20–1
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Example 20–2
Table 20–4
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Example 20–2
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Probability Distributions

Some standard discrete and continuous probability distributions
◦ Gaussian, or normal
◦ Lognormal
◦ Uniform
◦ Weibull
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Gaussian (Normal) Distribution

The Gaussian, or normal, distribution is defined as

It can be expressed as

Typical plots of the Gaussian distribution, with small or large
standard deviation, look like the following
Fig. 20–5
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Transformation Variate z

The deviation from the mean is expressed in units of standard
deviation by the transform

The transformation variate z is normally distributed, with a mean
of zero and a standard deviation and variance of unity.
That is, z = N (0, 1)

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Cumulative Distribution Function for Gaussian Distribution
Integration of the Gaussian distribution to find the cumulative
density function F(x) is accomplished numerically.
 To avoid the need for many tables for different values of mean and
standard deviation, the z transform is used.
 The integral of the transform is tabulated in Table A–10.
 A sketch of the standard normal distribution, showing the z
transform is given below.
 The normal cumulative density function is labeled F(z)

Fig. 20–6
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Example 20–3
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Example 20–3
Fig. 20–7
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Example 20–3
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Lognormal Distribution
In the lognormal distribution the logarithms of the variate have a
normal distribution.
 Thus the variate is said to by lognormally distributed.
 The parent, or principal, distribution is expressed as


The companion, or subsidiary, distribution is express through a
transformation,
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Lognormal Distribution
The probability density function (PDF) for x is derived from that
for y.
 The PDF for the companion distribution is

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Example 20–4
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Example 20–4
Table 20–5
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Example 20–5
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Example 20–5
Fig. 20–8
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Uniform Distribution

The uniform distribution is a closed-interval distribution that arises
when the chance of an observation is the same as the chance for
any other observation.
The probability density function (PDF) is

The cumulative density function, the integral of f(x), is

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Uniform Distribution

The mean for the uniform distribution,

The standard deviation,
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Weibull Distribution
The Weibull distribution is a flexible, asymmetrical distribution
with different values for mean and median.
 It contains within it a good approximation of the normal
distribution and an exact representation of the exponential
distribution.
 It is widely used to represent laboratory and field service data.

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Weibull Distribution

The reliability given by a three-parameter Weibull distribution is

For the special case when x0 = 0, the two-parameter Weibull is

If a specific reliability is given, solving Eq. (20–24) for x,
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Weibull Distribution
The characteristic variate q represents a value of x below which
lie 63.2% of the observations.
 The shape parameter b controls the skewness of the distribution.
 A good approximation to the normal distribution is obtained when
3.3 < b < 3.5
 The distribution is exponential when b = 1

Fig. 20–9
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Weibull Distribution

To find the probability function, note that
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Weibull Distribution

The mean and standard deviation are given by
G is the gamma function, tabulated in Table A–34.
 The notation for a Weibull distribution is

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Example 20–6
Shigley’s Mechanical Engineering Design
Example 20–6
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Propagation of Error

In the equation for axial stress

If F and A are random variables, then the stress is also a random
variable,

Errors inherent in F and A are said to be propagated to the stress
variate .
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Propagation of Error


Similar relations for other arithmetic functions are given in Table
20–6.
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Means and Standard Deviations for Algebraic Operations
Table 20–6
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Example 20-7
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Example 20-7
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Linear Regression
A process called regression is used to obtain a curve that best fits a
set of data points.
 The process is called linear regression when the best-fitting
straight line is found.

Fig. 20–10
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Linear Regression
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Linear Regression

The correlation coefficient r indicates the degree to which a set of
data points correlates with a regression line.

The standard deviations for the slope and intercept are given by
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Example 20–8
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Example 20–8
Table 20–7
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Example 20–8
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Example 20–8
Fig. 20–11
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