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The Lognormal Distribution

The Lognormal Distribution. By: Brian Shaw and Tim David. What is a Lognormal?. First Defined by McCallister(1879) A variation on the normal distribution Positively Skewed Used for things which have normal distributions with only positive values. Derivation of Lognormal.

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The Lognormal Distribution

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  1. The Lognormal Distribution By: Brian Shaw and Tim David

  2. What is a Lognormal? • First Defined by McCallister(1879) • A variation on the normal distribution • Positively Skewed • Used for things which have normal distributions with only positive values

  3. Derivation of Lognormal • A distribution whose logarithm is normally distributed • Let Y=lnX=v(y) • v’(y)=1/x • g(y)=[f(v(y))][v’(y)] • Similar to normal is defined for all positive values: 0≤x≤∞ Normal: Lognormal:

  4. Some Properties of Lognormal • C.D.F. • We could Integrate • That’s Hard • Make it Standard Normal

  5. Some Properties of Lognormal • C.D.F • Let’s Integrate: • Will be useful later • How do we do this? • Use the erf • Error function associated with normal distribution

  6. Some Properties of Lognormal • Normalization: • For computational simplicity we let y=ln(x) and differentiate • Then: dy=dx/x • Also realize that ey=x • This allows to write the lognormal as a normal distribution

  7. Some Properties of Lognormal • E(Y): • normally integrate f(x)x • Because we made substitution, integrate f(y)ey • Can you integrate this? • No!

  8. Some Properties of Lognormal • E(Y): • Because we can’t directly integrate use the erf • erf(∞)=1,erf(-∞)=-1

  9. Some Properties of Lognormal • Var(X) • Similarly we can computer the variance of a lognormal

  10. Skewness: A Defining Characteristic • The major difference between lognormal and normal distribution • Positive Domain • Positive Skewness

  11. Skewness: A Defining Characteristic • What is skewness? • A measure of asymmetry in a distribution • Defined by a Ratio: • Positive Ratio: • Long R.H. Tail • Negative ratio: • Long L.H. Tail • Ratio=0 • Symetric

  12. Skewness: A Defining Characteristic • For a lognormal distribution, skewness is defined by the formula:

  13. Mode: An old friend with a new take • Because the Lognormal distribution is skewed, the mean is not at the peak • This makes sense, because the tails are uneven.

  14. Mode: An old friend with a new take • Because of this problem we use the mode to describe the peak. • We can find the mode by maximizing the p.d.f • Do this by taking the derivative and setting it to zero

  15. Putting It All Together • Lets take a look at the first graph we had and apply some of things we learned. • Two Curves: • μ=5, σ2=.25(Blue) • Μ=5, σ2=2.25(Red)

  16. Putting It All Together Mode • Properties A Curve: • Blue • Mean: 168 • Variance: 8033 • Mode: 115 • Skewness:1.75 • Red • Mean:457 • Variance: 1773777 • Mode: 15 • Skewness: 33 Mean

  17. Which of these is not Lognormally Distributed? • Number of crystals in Ice Cream • Survival time after diagnosis in cancer • Age of marriage of women (Denmark) • Air pollution in PSI (Los Angeles) • Length of spoken words in phone conversation • Farm size in Wales(1989) • The height of St. Mary’s Students

  18. Armageddon! • Near Earth Asteroids(NEA’s) are an issue • The impact of a giant asteroid could end life as we know it! • Good thing we can calculate the probability of this happening.

  19. Armageddon! • Theory of Breakage • Get two rocks • Break them against each other • Examine the mass of the pieces • You can use lognormal to determine the mass of the nth generation • End up with 2n rocks

  20. Armageddon! • As you can see, this would also apply to asteroids. • They are just big rocks. • Traditionally, used the power law to describe the size of asteroids • Limitation: Couldn’t take single events into accounts. Theory of breakage allows us to do this • Therefore, we can use the lognormal!

  21. Armageddon! (With Math) • NEA’s come from fragmentation in the Main Asteroid Belt • From the data we have, we can model how many NEA’s are large enough to cause a significant problem. • This is because the crater is proportional to the asteroid size.

  22. Armageddon! (With Math) • Recently, estimations of NEA’s have shown that there are less significantly large NEA’s than once thought • This is supported by the math: • If we classify the asteroid size into two lognormal distributions, we get similar results.

  23. Concluding Remarks • Looks like the end of the world isn’t coming any time soon. • Study for finals.

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