1. Continuous Probability Distributions

2. 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

3. Expected Values of Continuous RVs

4. 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

5. 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
Navigation (+) 25 28 30
Recreation (+)
4200
5400
7800
Fish/Wildlife (+) 57
127
173
Area Redvlp (+)
830
1192
Capital Cost (-)
-193K
-180K
-162K
Annual Cost (-)
-7000
-6600
-6000
Operation/Maint(-)
-2192
-2049
-1742

6. 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(( )k) = 1.75k
Area above 12.0 is 0.5(( )k) = 1.50k
Total Area is 3.25k  k=1/3.25

7. Example – Cost/Benefit Analysis of Sprewell-Bluff Project (IV) (Flood Control)

8. Example – Cost/Benefit Analysis of Sprewell-Bluff Project (V) (Flood Control)

9. Example – Cost/Benefit Analysis of Sprewell-Bluff Project (VI) (Flood Control)

10. Example – Cost/Benefit Analysis of Sprewell-Bluff Project (VII) (Flood Control)

11. 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

12. Uniform Distribution - Expectations

13. 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

14. Exponential Density Functions (pdf)

15. Exponential Cumulative Distribution Functions (CDF)

16. Gamma Function
EXCEL Function: =EXP(GAMMALN(a))

17. Exponential Distribution - Expectations

18. Exponential Distribution - MGF

19. 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

20. 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)

21. Gamma/Exponential Densities (pdf)

22. Gamma Distribution - Expectations

23. Gamma Distribution - MGF

24. 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:

25. 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)

26. Normal Distribution – Density Functions (pdf)

27. Normal Distribution – Normalizing Constant

28. Obtaining Value of G(1/2)

29. Normal Distribution - Expectations

30. Normal Distribution - MGF

31. Normal(0,1) – Distribution of Z2

32. 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)

33. Beta Density Functions (pdf)

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

35. Weibull Density Functions (pdf)

36. Lognormal Distribution
Obtaining Probabilities in EXCEL:
To obtain: F(y)=P(Y≤y) Use Function: =LOGNORMDIST(y,m,s)

37. Lognormal pdf’s