Continuous Probability Distributions

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# Continuous Probability Distributions - PowerPoint PPT Presentation

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 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
• 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)

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
• 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
• 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

EXCEL Function: =EXP(GAMMALN(a))

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
• 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
• 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
• 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
• 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

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

Lognormal Distribution

Obtaining Probabilities in EXCEL:

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