Other Probability Density Functions

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Other Probability Density Functions
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PDF Applets

• Applets

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The Gamma Function
the gamma function
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Gamma Distribution
A continuous rv X has a gamma distribution if the
pdf is
where the parameters satisfy
The standard gamma distribution has
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Mean and Variance
The mean and variance of a random variable X
having the gamma distribution
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Probabilities from the Gamma Distribution
Let X have a gamma distribution with parameters

Then for any x gt 0, the cdf of X is given by
where
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Excel Functions

• GAMMADIST(x,alpha,beta,cumulative)
• GAMMAINV(probability,alpha,beta)

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Exponential Distribution
A continuous rv X has an exponential distribution
with parameter if the pdf is
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Mean and Variance
The mean and variance of a random variable X
having the exponential distribution
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Probabilities from the Exponential Distribution
Let X have a exponential distribution
Then the cdf of X is given by
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Excel Functions

• EXPONDIST(x,lambda,cumulative)

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Example
Suppose the response time X at a certain computer
terminal (the elapsed time between the end of a
users inquiry and the beginning of the systems
response to the inquiry) has an exponential
distribution with expected response time equal to
5 sec. What is the probability that the response
time is at most 10 seconds?
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The Chi-Squared Distribution
Let v be a positive integer. Then a random
variable X is said to have a chi-squared
distribution with parameter v if the pdf of X is
the gamma density with
The pdf is
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The Chi-Squared Distribution
The parameter v is called the number of degrees
of freedom (df) of X. The symbol is often
used in place of chi-squared.
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Excel Functions

• CHIDIST(x,deg_freedom)
• CHIINV(probability,deg_freedom)

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Weibull Distribution

• The Weibull distribution is one of the most
  widely used distributions in reliability
  engineering. It is a versatile distribution that
  can take on the characteristics of other types of
  distributions, based on the value of the shape
  parameter.
• http//www.itl.nist.gov/div898/handbook/eda/sectio
  n3/eda3668.htm

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The Weibull Distribution
A continuous rv X has a Weibull distribution if
the pdf is
where the parameters satisfy
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Mean and Variance
The mean and variance of a random variable X
having the Weibull distribution are
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Weibull Distribution
The cdf of a Weibull rv having parameters
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Excel Functions

• WEIBULL(x,alpha,beta,cumulative)

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Lognormal Distribution
A nonnegative rv X has a lognormal distribution
if the rv Y ln(X) has a normal distribution the
resulting pdf has parameters
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Mean and Variance
The mean and variance of a variable X having the
lognormal distribution are
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Lognormal Distribution
The cdf of the lognormal distribution is given by
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Excel Functions

• LOGNORMDIST(x,mean,standard_dev)
• LOGINV(probability,mean,standard_dev)

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Beta Distribution
A rv X is said to have a beta distribution with
parameters A, B,
if the pdf of X is
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Comments

• The beta distribution describes a density
  function over an interval of finite length.
• Models variation in the proportion or percentage
  of a quantity occurring in different samples.

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Mean and Variance
The mean and variance of a variable X having the
beta distribution are
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Excel Functions

• BETADIST(x,alpha,beta,A,B)
• BETAINV(probability,alpha,beta,A,B)

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Probability Plots

• The probability plot is a graphical technique for
  assessing whether or not a data set follows a
  given distribution such as the normal or Weibull.
• The data are plotted against a theoretical
  distribution in such a way that the points should
  form approximately a straight line. Departures
  from this straight line indicate departures from
  the specified distribution.

http//www.itl.nist.gov/div898/handbook/eda/sectio
n3/probplot.htm
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Sample Percentile
Order the n-sample observations from smallest to
largest. The ith smallest observation in the
list is taken to be the 100(i 0.5)/nth sample
percentile.
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Probability Plot
If the sample percentiles are close to the
corresponding population distribution
percentiles, the first number will roughly equal
the second.
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Normal Probability Plot
A plot of the pairs
On a two-dimensional coordinate system is called
a normal probability plot. If the drawn from a
normal distribution the points should fall close
to a line with slope and intercept
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Example
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P-P in SPSS

• Exercise 88, p. 178

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Beyond Normality
Consider a family of probability distributions
involving two parameters
Let denote the
corresponding cdfs. The parameters
are said to location and scale
parameters if