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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. - PowerPoint PPT Presentation
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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
• 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:€
12πσ
e−(x−μ )2
2σ 2
€
1x 2πσ
e−(ln(x )−μ )2
2σ 2
Some Properties of Lognormal
• C.D.F.• We could Integrate
• That’s Hard • Make it Standard
Normal
€
φ ln x −μσ
⎛ ⎝ ⎜
⎞ ⎠ ⎟€
12πσ
1te
−(ln( t )−μ )2
2σ 2
0
x
∫ dt
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
€
12πσ
1te
−(ln( t )−μ )2
2σ 2
0
x
∫ dt
€
F(x) = 12
1+ erf ln x −μσ 2
⎛ ⎝ ⎜
⎞ ⎠ ⎟
⎡ ⎣ ⎢
⎤ ⎦ ⎥€
erf (x) =2π
e−t 2
dt0
x
∫
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
€
12πσ
1xe
−(ln(x )−μ )2
2σ 2
0
∞
∫ dx =1
€
12πσ
e−(y−μ )2
2σ 2
−∞
∞
∫ dy =1
Some Properties of Lognormal
• E(Y): • normally integrate
f(x)x • Because we made
substitution, integrate f(y)ey
• Can you integrate this?• No!
€
E(X) = 12πσ
e−(y−μ )2
2σ 2
−∞
∞
∫ eydy
Some Properties of Lognormal
• E(Y):• Because we can’t
directly integrate use the erf
• erf(∞)=1,erf(-∞)=-1
€
E(X) = 12πσ
e−(y−μ )2
2σ 2
−∞
∞
∫ eydy
€
E(X) = − 12eμ +σ 2
2 erf μ +σ 2 − y2σ
⎛ ⎝ ⎜
⎞ ⎠ ⎟−∞
∞
€
E(X) = − 12eμ +σ 2
2 (−1) −1[ ] = − 12
(−2)eμ +σ 2
2 = eμ +σ 2
2
Some Properties of Lognormal
• Var(X)• Similarly we can
computer the variance of a lognormal
€
Var (X) = E(X 2) −μ 2
= 12πσ
e−(y−μ )2
2σ 2
−∞
∞
∫ e2ydy − eμ +σ 2
2 ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
2
= eσ2 +2μ (eσ
2
−1)
Skewness: A Defining Characteristic
• The major difference between lognormal and normal distribution• Positive Domain• Positive Skewness
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
€
γ1 = μ3
μ23 / 2
Skewness: A Defining Characteristic
• For a lognormal distribution, skewness is defined by the formula:
€
eσ2
−1 2 + eσ2
( )
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.
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
€
f (x) = 1x 2πσ
e−(ln(x )−μ )2
2σ 2
€
f '(x) = 0
€
Mode(X ) = eμ−σ 2
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)
Putting It All Together
• Properties A Curve:• Blue
• Mean: 168• Variance: 8033• Mode: 115• Skewness:1.75
• Red• Mean:457• Variance: 1773777• Mode: 15• Skewness: 33
Mode
Mean
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
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.
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
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!
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.
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.