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Special Continuous Probability Distribution Lognormal Distribution

PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS. Special Continuous Probability Distribution Lognormal Distribution. f(x). x. 0. Lognormal Distribution – Probability Density Function.

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Special Continuous Probability Distribution Lognormal Distribution

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  1. PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS Special Continuous Probability DistributionLognormal Distribution

  2. f(x) x 0 Lognormal Distribution – Probability Density Function A random variable X is said to have the Lognormal Distribution with parameters  and , where  > 0 and  > 0, if the probability density function of X is: , for X >0 , for X 0 

  3. Lognormal Distribution If X ~ LN(,), then Y= ln (X) ~ N(,)

  4. Lognormal Distribution - Probability Distribution Function where F(z) is the cumulative probability distribution function of N(0,1)

  5. Lognormal Distribution Mean or Expected Value of X Percentile of X Standard Deviation of X

  6. 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 µ=5 and σ=0.1 , Find: µx and σx P(X >120) 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.

  7. Lognormal Distribution –Example Solution (a)

  8. Lognormal Distribution –Example Solution (b)

  9. Lognormal Distribution –Example Solution (C) (d)

  10. Lognormal Distribution –Example Solution (e) Let Y=number of items tested that have strength of at least 120 y=0,1,2,…,10

  11. Lognormal Distribution –Example Solution f) The value of x, say xms, for which is determined as follows: and , , so that , therefore

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