Continuous Probability Distributions

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Continuous Probability Distributions
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Continuous Random Variables and Probability
Distributions

• Random Variable Y
• Cumulative Distribution Function (CDF)
  F(y)P(Yy)
• Probability Density Function (pdf) f(y)dF(y)/dy
• Rules governing continuous distributions
• f(y) 0 ? y
• P(aYb) F(b)-F(a)
• P(Ya) 0 ? a

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Expected Values of Continuous RVs
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Example Cost/Benefit Analysis of Sprewell-Bluff
Project (I)

• Subjective Analysis of Annual Benefits/Costs of
  Project (U.S. Army Corps of Engineers
  assessments)
• Y Actual Benefit is Random Variable taken from
  a triangular distribution with 3 parameters
• ALower Bound (Pessimistic Outcome)
• BPeak (Most Likely Outcome)
• CUpper Bound (Optimistic Outcome)
• 6 Benefit Variables
• 3 Cost Variables

Source B.W. Taylor, R.M. North(1976). The
Measurement of Uncertainty in Public Water
Resource Development, American Journal of
Agricultural Economics, Vol. 58, 4, Pt.1,
pp.636-643
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Example Cost/Benefit Analysis of Sprewell-Bluff
Project (II) (1000s, rounded)
Benefit/Cost Pessimistic (A) Most Likely (B) Optimistic (C)
Flood Control () 850 1200 1500
Hydroelec Pwr () 5000 6000 6000
Navigation () 25 28 30
Recreation () 4200 5400 7800
Fish/Wildlife () 57 127 173
Area Redvlp () 0 830 1192
Capital Cost (-) -193K -180K -162K
Annual Cost (-) -7000 -6600 -6000
Operation/Maint(-) -2192 -2049 -1742
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Example Cost/Benefit Analysis of Sprewell-Bluff
Project (III) (Flood Control, in 100K)
Triangular Distribution with lower bound8.5
Peak12.0 upper bound15.0
Choose k ? area under density curve is 1 Area
below 12.0 is 0.5((12.0-8.5)k) 1.75k Area
above 12.0 is 0.5((15.0-12.0)k) 1.50k Total
Area is 3.25k ? k1/3.25
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Example Cost/Benefit Analysis of Sprewell-Bluff
Project (IV) (Flood Control)
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Example Cost/Benefit Analysis of Sprewell-Bluff
Project (V) (Flood Control)
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Example Cost/Benefit Analysis of Sprewell-Bluff
Project (VI) (Flood Control)
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Example Cost/Benefit Analysis of Sprewell-Bluff
Project (VII) (Flood Control)
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Uniform Distribution

• Used to model random variables that tend to occur
  evenly over a range of values
• Probability of any interval of values
  proportional to its width
• Used to generate (simulate) random variables from
  virtually any distribution
• Used as non-informative prior in many Bayesian
  analyses

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Uniform Distribution - Expectations
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Exponential Distribution

• Right-Skewed distribution with maximum at y0
• Random variable can only take on positive values
• Used to model inter-arrival times/distances for a
  Poisson process

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Exponential Density Functions (pdf)
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Exponential Cumulative Distribution Functions
(CDF)
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Gamma Function
EXCEL Function EXP(GAMMALN(a))
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Exponential Distribution - Expectations
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Exponential Distribution - MGF
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Exponential/Poisson Connection

• Consider a Poisson process with random variable X
  being the number of occurences of an event in a
  fixed time/space X(t)Poisson(lt)
• Let Y be the distance in time/space between two
  such events
• Then if Y gt y, no events have occurred in the
  space of y

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Gamma Distribution

• Family of Right-Skewed Distributions
• Random Variable can take on positive values only
• Used to model many biological and economic
  characteristics
• Can take on many different shapes to match
  empirical data

Obtaining Probabilities in EXCEL To obtain
F(y)P(Yy) Use Function
GAMMADIST(y,a,b,1)
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Gamma/Exponential Densities (pdf)
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Gamma Distribution - Expectations
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Gamma Distribution - MGF
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Gamma Distribution Special Cases

• Exponential Distribution a1
• Chi-Square Distribution an/2, b2 (n
  integer)
• E(Y)n V(Y)2n
• M(t)(1-2t)-n/2
• Distribution is widely used for statistical
  inference
• Notation Chi-Square with n degrees of freedom

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Normal (Gaussian) Distribution

• Bell-shaped distribution with tendency for
  individuals to clump around the group median/mean
• Used to model many biological phenomena
• Many estimators have approximate normal sampling
  distributions (see Central Limit Theorem)

Obtaining Probabilities in EXCEL To obtain
F(y)P(Yy) Use Function
NORMDIST(y,m,s,1)
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Normal Distribution Density Functions (pdf)
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Normal Distribution Normalizing Constant
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Obtaining Value of G(1/2)
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Normal Distribution - Expectations
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Normal Distribution - MGF
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Normal(0,1) Distribution of Z2
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Beta Distribution

• Used to model probabilities (can be generalized
  to any finite, positive range)
• Parameters allow a wide range of shapes to model
  empirical data

Obtaining Probabilities in EXCEL To obtain
F(y)P(Yy) Use Function
BETADIST(y,a,b)
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Beta Density Functions (pdf)
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Weibull Distribution
Note The EXCEL function WEIBULL(y,a,b) uses
parameterization ab, bab
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Weibull Density Functions (pdf)
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Lognormal Distribution
Obtaining Probabilities in EXCEL To obtain
F(y)P(Yy) Use Function
LOGNORMDIST(y,m,s)
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Lognormal pdfs