Asymmetric distributions

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LECTURE 8

• Asymmetric distributions
• Lognormal, exponential, Weibull
• Fitting the above densities to data
• Probability computations from densities
• Reducing to a standard density computation
• Normal distribution computations
• Log-normal distribution computations
• Using S-PLUS functions such as
• pnorm,plogis,pexp,

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pdf for log-normal data X
Not quite the same form as a normal density!
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Log-normal distribution
Useful for data that takes on only positive
values, and for which the log transformation
produces approximately normally distributed data
xlnorm rlnorm(200, 2, 1) log-normal data
for histogram
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pdf for Y ln(X)
A normal distribution pdf
NOTE The above result requires a mathematical
proof that is beyond the scope of our present
emphasis.
THE NICE THING If you have data
that has a log-normal
distribution, then the transformed data

has a normal distribution
with parameters . So fit a normal
density to the
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First make the log transform, and fit the normal
density

• gt log.xlnorm log(xlnorm)
• gt mu mean(log.xlnorm)
• gt sd stdev(log.xlnorm)
• gt hist(log.xlnorm,
• probability T, col 0)
• gt x seq(-1, 5, 0.1)
• gt lines(x, dnorm(x, mu, sd))
• gt mu
• 1 2.037
• gt sd
• 1 1.012
• gt title(main "HISTOGRAM

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Then fit the log-normal density
hist(xlnorm,probabilityT,col0,xlimc(0,120),ylim
c(0,.09)) mumean(log.xlnorm) sdstdev(log.xlnorm
) xseq(0,120,.1) lines(x,dlnorm(x,mu,sd))
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The Weibull distribution

• A generalization of the exponential distribution
  that is particularly useful in reliability
  modeling applications
• Show and discuss material on pages 44-45 of
  Devore and Farnum
• Fitting a Weibull density to data
• Requires more tools than we currently have at our
  fingertips.
• But we can easily fit an exponential density.
  See next slide(s)

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Fitting an exponential density to data
From Lecture 7 (see handout, page 7.4), we know
that the p-quantile x(p) for an exponential
density is given by the expression
Textbook notation
For p .5 we have the expression for the median
of the density
Given you can compute
, OR Given
you can compute
!
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Now it is easy to do the fit
We see that
So just compute the sample median
and Substitute it in place of the median of
the density
to get
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Computing Probs. from Densities A Key
Location and scale family of densities
Change of variable
Standard form of density
Thats all there is to the tricky part of it!
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You also use
Draw sketches
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Normal density calculations andS-PLUS functions

• Go over some normal density calculations
• Section 1.4 Devore and Farnum
• Can use Table 1 or S-PLUS function pnorm
• Highly recommend reading pp, 55-63, S-PLUS 6
  Guide to Statistics, Vol. 1 (on-line)
• Use of pnorm, qnorm, dnorm, rnorm, and similarly
  for other distributions.