Special Continuous Probability Distribution Lognormal Distribution

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Special Continuous Probability DistributionLogn
ormal Distribution

• PROBABILITY AND STATISTICS FOR SCIENTISTS AND
  ENGINEERS

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Lognormal Distribution Probability Density
Function

• A random variable X is said to have the Lognormal
  Distribution with parameters ? and ?, where ? gt 0
  and ? gt 0, if the probability density function of
  X is
• 
  , for X
  gt0
• 
  , for X
  0
• 
  

?
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Lognormal Distribution

• If X LN(?,?),
• then Y ln (X) N(?,?)

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Lognormal Distribution - Probability Distribution
Function

• where F(z) is the cumulative probability
  distribution function of N(0,1)

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Lognormal Distribution

• Mean or Expected Value of X
• Percentile of X
• Standard Deviation of X

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Lognormal Distribution - Example

• A theoretical justification based on a certain
  material failure mechanism underlies the
  assumption that ductile strength X of a material
  has a lognormal distribution.
• If the parameters are µ5 and s0.1 ,
• Find
• µx and sx
• P(X gt120)
• P(110 X 130)
• The median ductile strength
• The expected number having strength at least 120,
  if ten different samples of an alloy steel of
  this type were subjected to a strength test.
• (f) The minimum acceptable strength, If the
  smallest 5 of strength values were unacceptable.

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Lognormal Distribution Example Solution

• (a)

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Lognormal Distribution Example Solution

• (b)

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Lognormal Distribution Example Solution

• (C)
• (d)

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Lognormal Distribution Example Solution

• (e) Let Ynumber of items tested that have
  strength of at
• least 120 y0,1,2,,10

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Lognormal Distribution Example Solution

• f) The value of x, say xms, for which
  is
• determined as follows
• 
  
• and
  ,
• 
  ,
• so that
  
• 
  ,
• therefore