Distributions (3) © 2008 Winton 2 VIII

Home , Beta distribution, Cauchy distribution, Normal distribution

• Data points t are said to be lognormally distributed if the natural logarithms, ln(t), of these points are normally distributed with mean μ and standard deviation σ – If the normal distribution is sampled to get points rsample, then the points ersample constitute sample values from the lognormal distribution • The pdf for the lognormal distribution is given by ()ln(x)- - μ 2 1 2 f(x) ⋅= e 2σ (x > 0) x 2πσ because ∞ ()ln(x)- - μ 2 ∞ -() -t μ 2 1 1 2 1 2 ∫ ⋅e 2σ dx ∫ ⋅= e 2σ (wheredt t = ln(x)) 0 x 2πσ 0 2πσ is the pdf for the normal distribution © 2008 Winton 3 Mean and Variance for Lognormal • It can be shown that in terms of μ and σ σ2 μ + e E(X) = e 2

2 2 1 - ee Var(X) 2 μ +⋅ ⋅= (ee σσ - 1 ) • If the mean E and variance V for the lognormal distribution are given, then the corresponding μ and σ2 for the normal distribution are given by – σ2 = log(1+V/E2) – μ = log(E) - σ2/2 • The lognormal distribution has been used in reliability models for time until failure and for stock price distributions – The shape is similar to that of the Gamma distribution and the Weibull distribution for the case α > 2, but the peak is less towards 0 © 2008 Winton 4 Graph of Lognormal pdf

f(x) 0.5 0.45 μ=4, σ=1 0.4 0.35 μ=4, σ=2 0.3 μ=4, σ=3 0.25 μ=4, σ=4 0.2 0.15 0.1 0.05

• Beta function – a function studied by Euler prior to his work on the gamma function given by -1 ()()ΓΓ βα ) , B(α , β) = α - 1 ()1x x- β - 1 dx B(β , α) == ∫ (α +Γ β) 0+ • For 0 < x < 1 and α and β the beta distribution is given by the pdf

• It can be shown that α E(X) = α + β αβ Var(X) = ()(α 2 αβ β ++⋅+ 1 ) • If α > 1 and β > 1 –Mode(X)=(α -1)/(α + β -2) – Otherwise the mode is an end point or is not unique • α and β serve as shape parameters for the distribution, which takes on very different shapes on the (0,1) interval to which it is restricted – Note that if α = β = 1, then f(x) = 1 and the distribution is just the uniform distribution for (0,1) © 2008 Winton 7 Beta Distribution Uses • Sometimes used as a rough model in the absence of data as an alternative to the triangular distribution – If there are estimates for minimum a, most likely c, and maximum b, and the mean μ is also estimated, then α and β can in turn be estimated from μ = a + α(b - a)/(α + β) c = a + (α - 1)(b - a)/(α + β -2) – Solve for α and β • The 2 equations for (E(X) and Mode(X)) have been adjusted to correspond to the interval (a,b) rather than (0,1) • The distribution has also been used to estimate the number of defective items in a collection or the time to complete a task © 2008 Winton 8

Graphing Beta Distribution pdf

• For the Beta distribution pdf ⎧ if α <∞ 1 ⎧ if β <∞ 1 ⎪ ⎪ f(x) lim f(x) = ⎨ β if α = 1 and lim f(x) = ⎨α if β = 1 →0x ⎪ x→ 1 ⎪ ⎩ 0 if α > 1 ⎩0 if β > 1 • This facilitates determining the manner in which α and β govern the appearance of the distribution as graphed © 2008 Winton 9

Configurations α > β, α > 1 (reversing α and β flips the graph left-right)

f(x)

5 α=5, β=0.5

4 α=5, β=1

3 α=5, β=1.5

2 α=5, β=2 α=5, β=2.5 1

Configurations α = β f(x)

3 α=1.5, β=1.5 α=2, β=2 α=0.5, β=0.5 α=2.5, β=2.5 2 α=1, β=1

Configurations α < β, α < 1

f(x)

α=1, β=2 α=.9, β=4 2 α=.8, β=8 α=.7, β=16

• There are many other distributions that have been devised • Partial list (Red indicates implemented in Extend): – Cauchy Distribution (derived from the Normal Distribution) – Chi-squared Distribution (the Gamma Distribution with α=r/2 for r an integer) – Dirichlet (the multivariate generalization of the Beta Distribution) – F Distribution (extension of Chi-squared Distribution) – Hyperexponential Distribution (not analogous to the hypergeometric distribution) – Hypergeometric Distribution (related to the binomial distribution) – Laplace Distribution (combination of 2 exponential distributions for a spike) – Logistic Distribution (like the Normal, but with heavier tails) – Multinomial Distribution (generalization of the Binomial Distribution) – Negative Binomial Distribution (generalization of the Geometric Distribution ) – Pareto Distribution (related to the Exponential Distribution) – T Distribution (Student’s t-distribution used for T tests where Normal doesn’t apply) – Wald (Inverse Gaussian) Distribution (related to the Normal Distribution [which is sometimes called the Gaussian Distribution] but the term inverse is not in the mathematical sense)