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Continuous Probability Distributions. Continuous Random Variables and Probability Distributions. Random Variable: Y Cumulative Distribution Function (CDF): F ( y )=P( Y ≤ y ) Probability Density Function (pdf): f ( y )=d F ( y )/d y Rules governing continuous distributions:

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continuous random variables and probability distributions
Continuous Random Variables and Probability Distributions
  • Random Variable: Y
  • Cumulative Distribution Function (CDF): F(y)=P(Y≤y)
  • Probability Density Function (pdf): f(y)=dF(y)/dy
  • Rules governing continuous distributions:
    • f(y) ≥ 0  y
    • P(a≤Y≤b) = F(b)-F(a) =
    • P(Y=a) = 0  a
example cost benefit analysis of sprewell bluff project i
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:
    • A=Lower Bound (Pessimistic Outcome)
    • B=Peak (Most Likely Outcome)
    • C=Upper 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

example cost benefit analysis of sprewell bluff project iii flood control in 100k
Example – Cost/Benefit Analysis of Sprewell-Bluff Project (III)(Flood Control, in $100K)

Triangular Distribution with:

lower bound=8.5

Peak=12.0

upper bound=15.0

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 k=1/3.25

uniform distribution
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
exponential distribution
Exponential Distribution
  • Right-Skewed distribution with maximum at y=0
  • Random variable can only take on positive values
  • Used to model inter-arrival times/distances for a Poisson process
gamma function
Gamma Function

EXCEL Function: =EXP(GAMMALN(a))

exponential poisson connection
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 > y, no events have occurred in the space of y
gamma distribution
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(Y≤y) Use Function: =GAMMADIST(y,a,b,1)

gamma distribution special cases
Gamma Distribution – Special Cases
  • Exponential Distribution – a=1
  • Chi-Square Distribution – a=n/2, b=2 (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:
normal gaussian distribution
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(Y≤y) Use Function: =NORMDIST(y,m,s,1)

beta distribution
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(Y≤y) Use Function: =BETADIST(y,a,b)

weibull distribution
Weibull Distribution

Note: The EXCEL function WEIBULL(y,a*,b*) uses parameterization: a*=b, b*=ab

lognormal distribution
Lognormal Distribution

Obtaining Probabilities in EXCEL:

To obtain: F(y)=P(Y≤y) Use Function: =LOGNORMDIST(y,m,s)