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Continuous Probability Distributions (PPT)

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Page 1: Continuous Probability Distributions (PPT)

Continuous Probability Distributions

Page 2: Continuous Probability Distributions (PPT)

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

b

adyyf )(

1)(

dyyf

Page 3: Continuous Probability Distributions (PPT)

Expected Values of Continuous RVs

a

aYVadyyfyadyyfaay

dyyfbabaybaYEbaYEbaYV

baba

dyyfbdyyyfadyyfbaybaYE

YEYE

dyyfdyyyfdyyfydyyfyy

dyyfyYEYEYV

dyyfygYgE

dyyyfYE

baY

222222

22

2222

2222

222

)()()()()(

)()()()()(

)1()(

)()()()(

)1()(2

)()(2)()(2

)()())(()( :Variance

)()()(

e)convergenc absolute (assuming )()( :Value Expected

Page 4: Continuous Probability Distributions (PPT)

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

Page 5: Continuous Probability Distributions (PPT)

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

Page 6: Continuous Probability Distributions (PPT)

Example – Cost/Benefit Analysis of Sprewell-Bluff Project (III) (Flood Control, in $100K)

)0.15,5.8( elsewhere 0

0.150.120.3/)0.15(0.125.85.3)5.8(

)(yy

yykyyk

yf

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

Triangular Distribution (Not Scaled)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 15.5 16

Flood Control Benefits ($100K)

Prob

abili

ty D

ensi

ty

Page 7: Continuous Probability Distributions (PPT)

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

2

8.5 8.5

2 2 2 2

( 8.5) 11.375 8.5 12.0( ) (15.0 ) / 9.75 12.0 15.0

0 elsewhere

8.5 ( ) 0

8.5 12 ( ) ( 8.50) 11.375 (1/11.375) 2 8.5

2 8.5 8.5 2 8.5 11.375 17

yy

y yf y y y

y F y

y F y t dt t t

y y y y

2

2

12 12

2 2

2

8.5 22.75

12 15 ( ) (12) (15 ) 9.75 .5385 (1/ 9.75) 15 2

.5385 15 2 15(12) 12 2 9.75

.5385 216 30 19.5 12 15

15 ( ) 1

yyy F y F t dt t t

y y

y y y

y F y

Page 8: Continuous Probability Distributions (PPT)

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

15 11512051282.0538462.1538462.10125.8043956.0747253.0175824.3

5.8 0

)(

elsewhere 00.150.1275.9/)0.15(0.125.8375.11)5.8(

)(

2

2

yyyyyyy

y

yF

yyyy

yf

Page 9: Continuous Probability Distributions (PPT)

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

Cumulative Distribution Function

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 15.5 16

y

F(y)

Page 10: Continuous Probability Distributions (PPT)

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

01.102.1

02.119.14021.14184.1121.141)()(

21.14182.30388.37061.4454.2974.45556.75964.111252.148346.17885.13315.50269.531

5.4512

125.34)12(15

5.4515

125.3415

25.295.8

395.8

25.29)12(5.8

3912

5.45125.3415

25.295.8

39375.1115

75.95.8)(

84.1141.4445.4949.1069.364.5095.9490.9835.14849.3100.2177.6208.59

125.3412

75.22)12(15

125.3415

75.2215

5.195.8

25.295.8

5.19)12(5.8

25.2912

125.3475.2215

5.195.8

25.29375.1115

75.95.8)()(

222

43444434

15

12

4312

5.8

3415

12

212

5.8

222

32333323

15

12

3212

5.8

2315

12

12

5.8

YEYEYV

yyyydyyydyyydyyfyYE

yyyydyyydyyydyyyfYE

Page 11: Continuous Probability Distributions (PPT)

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

elsewhere 0

1

)(bya

abyf

by

byaabay

ay

yF

1

0

)(

Page 12: Continuous Probability Distributions (PPT)

Uniform Distribution - Expectations

)(2887.01212

)(12

)(12

212

)2(3)(423

)()()(

3)(

)(3))((

)(3311

2)(2))((

)(2211)(

2

2222222

22222

22

2233322

222

ababab

ababbaabababba

ababbaYEYEYV

abba

ababbaab

ababy

abdy

abyYE

abababab

ababy

abdy

abyYE

b

a

b

a

b

a

b

a

Page 13: Continuous Probability Distributions (PPT)

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

elsewhere0

01

)(

/ ye

yf

y

011111)( 0

00

yeeeedteyF yyytty

Page 14: Continuous Probability Distributions (PPT)

Exponential Density Functions (pdf)

Exponential pdf's

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7 8 9 10

y

f(y)

f(y|th=1)f(y|th=2)f(y|th=5)f(y|th=10)

Page 15: Continuous Probability Distributions (PPT)

Exponential Cumulative Distribution Functions (CDF)

Exponential CDF

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 3 6 9 12 15

y

F(y)

F(y|th=1)F(y|th=2)F(y|th=5)F(y|th=10)

Page 16: Continuous Probability Distributions (PPT)

Gamma Function

)()(

: Letting :integral heConsider t

)!1()( integer,an is if that Note

Property) (Recursive )()0(0

)1(

:Partsby gIntegratin )1(

)(

0

1

0

1

0

1

0

1

0

1

0

1

00

1

0

0

1

dxexdxexdyey

dxdyxy

yxdyey

dyey

dyeyeyvduuvdyey

evdyedv

dyyduyu

dyey

dyey

xxy

y

y

yyy

yy

y

y

EXCEL Function: =EXP(GAMMALN(

Page 17: Continuous Probability Distributions (PPT)

Exponential Distribution - Expectations

22222

223

0

13

0

2

0

22

2

0

12

00

)(2)()(

2)!13()3(1

11

)!12()2(1

11)(

YEYEYV

dyeydyeydyeyYE

dyeydyyedyeyYE

yyy

yyy

Page 18: Continuous Probability Distributions (PPT)

Exponential Distribution - MGF

2222

323

22

1**

0

**

*

0

*

0

1

0

1

0

2)0(')0('')(

)0(')(

)1(2)()1(2)(''

)1()()1(1)('

)1(1

1)10(111)(

1 where11

11)(

MMYV

MYE

tttM

tttM

tt

etM

tdyedye

dyedyeeeEtM

y

yty

tyytytY

Page 19: Continuous Probability Distributions (PPT)

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(t)

• 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

1mean with lExponentia are ProcessPoisson in distances arrivals-Inter 1!0

)()0)(( :yProbabilitPoisson

)( :" Survival" lExponentia0

yy

y

eyeyXP

eyYP

Page 20: Continuous Probability Distributions (PPT)

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

otherwise 0

0,,0)(

1

)(

1

yey

yf

y

Obtaining Probabilities in EXCEL:

To obtain: F(y)=P(Y≤y) Use Function: =GAMMADIST(y,,1)

Page 21: Continuous Probability Distributions (PPT)

Gamma/Exponential Densities (pdf)Exponential and Gamma density functions

0

0.1

0.2

0.3

0.4

0.5

0 2 4 6 8 10

y

f(y)

exp(2.0)

exp(5.0)

gam(2,2)

gam(2,3)

gam(3,2)

Page 22: Continuous Probability Distributions (PPT)

Gamma Distribution - Expectations

222222222

22

22

0

1)2(

0

1

0

122

1

0

1)1(

00

1

)()1()()(

)1()(

)()1()(

)1()1(

)()2()2(

)(1

)(1

)(1

)(1

)()(

)()1()1(

)(1

)(1

)(1

)(1)(

YEYEYV

dyey

dyeydyeyyYE

dyey

dyeydyeyyYE

y

yy

y

yy

Page 23: Continuous Probability Distributions (PPT)

Gamma Distribution - MGF

2222

222

11

*

*

0

*1

0

11

0

11

0

1

)()1()0(')0('')(

)0(')(

)1()1()()1()1()(''

)1()()1()('

)1()()(

1)(

1 where

)(1

)(1

)(1

)(1)(

MMYV

MYE

tttM

tttM

ttM

tdyey

dyeydyey

dyeyeeEtM

y

tyty

ytytY

Page 24: Continuous Probability Distributions (PPT)

Gamma Distribution – Special Cases

• Exponential Distribution –

• Chi-Square Distribution – (≡ integer)– E(Y)= V(Y)=2– M(t)=(1-2t)-

– Distribution is widely used for statistical inference– Notation: Chi-Square with degrees of freedom:

2~ Y

Page 25: Continuous Probability Distributions (PPT)

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)

0,,2

1)( 2

2)(21

2

yeyfy

Obtaining Probabilities in EXCEL:To obtain: F(y)=P(Y≤y) Use Function: =NORMDIST(y,,,1)

Page 26: Continuous Probability Distributions (PPT)

Normal Distribution – Density Functions (pdf)

Normal Densities

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0 20 40 60 80 100 120 140 160 180 200

y

f(y)

N(100,400)

N(100,100)

N(100,900)

N(75,400)

N(125,400)

Page 27: Continuous Probability Distributions (PPT)

Normal Distribution – Normalizing Constant

2222

2

0

2

0

2

0

2

00

21

222

0 0212

0 0

sincos21

21212

2121

2121

22

12

2

222)(

2)(

222

2))1(0(

)1sin(cos

and )2,0[),,0( :domains with sin,cos :Ordinates-CoPolar toChanging

1 : variablesChanging

)for solve want to(we :integral heConsider t

2

222222

21

22

21

22

21

22

2

2

2

2

kkk

ddde

rdrderdrdedzdzek

rdrddzdzrrzrz

dzdzedzedzek

dzekdzedyek

dzdydydzyz

kkdye

r

r

rrzz

zzzz

zzy

y

Page 28: Continuous Probability Distributions (PPT)

Obtaining Value of

22122

1222

1

2 :Variables Changing

21 :Consider Now,

221

2 :get weslide, Previous From

0

2

0

2

0

2212

2

0

21

0

121

0

2

2

2

22

2

2

dze

zdzez

zdzez

zdzduzu

dueudueu

dze

dze

z

zz

uu

z

z

Page 29: Continuous Probability Distributions (PPT)

Normal Distribution - Expectations

222

2

222

232/323

0

2123

0

2

021

21

0

2

2

21

0

221

22

21

21

21

21

)1()()()(

)0()()()( then ,~ If :Note

1101)()(

122

2221

21

212

23

21

21

21

22

22

212

21 2 :Variables Changing

212

21

0)0(021

21

21)(

21)()1,0(~

22

22

222

2

ZVZVYV

ZEZEYEZYNY

ZEZEZV

dueu

dueuzdzzedzez

zdzduzdzduzu

dzezdzezZE

edzezdzezZE

ezfNZ

u

uzz

zz

zzz

z

Page 30: Continuous Probability Distributions (PPT)

Normal Distribution - MGF

2exp

22exp)(

,~ :R.V. normal a ofdensity over the gintegratin isit since 1, being integrallast The

)(21exp

21

22exp

22)(

21exp

21

22

22)(

2exp

21

22

22

2)(

2exp

21)(

2)( :square theCompleting

2

)(2

exp2

1

22exp

21

21exp

21)(

22

2

222

22

2

22

22

222

2

222

2

22

2

2

222

2

2222

2

2

2

2

2

2

222

2

222

2

2

2

2

2

2

2

222222

2

2

2

2

2

2

2

2

2

22

2

22

2

2

tttttM

tNY

dytytt

dyttty

dytttttyy

dytttttyytM

ttt

dytyy

dytyyydyyeeEtM tytY

Page 31: Continuous Probability Distributions (PPT)

Normal(0,1) – Distribution of Z2

21

2

212121

0

212

121

0

212

2

0

212

0

221

21

2

~

)21()21(2

221

221

21

21

212)(

21 and :Variables Changing

222

21

21)(

2

22

2222

2

Z

ttt

dueuduu

etM

duu

dzuzzu

dzedze

dzedzeeeEtM

tu

tu

Z

tztz

tzztztZ

Z

Page 32: Continuous Probability Distributions (PPT)

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

otherwise 0

0,,10)1()()()(

)(

11 yyy

yf

Obtaining Probabilities in EXCEL:To obtain: F(y)=P(Y≤y) Use Function: =BETADIST(y,a,b)

Page 33: Continuous Probability Distributions (PPT)

Beta Density Functions (pdf)Beta Density Functions

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

y

f(y)

Beta(1,1)

Beta(2,2)

Beta(4,1)

Beta(1,3)

Beta(5,5)

Page 34: Continuous Probability Distributions (PPT)

Weibull Distribution

1121)()(

21)(exp

11)(exp)(

)( and : variablesChanging

exp)(

0for expexp)()(

)0,(0exp1

00)(

2222

2

0

22

0

2

0

122

1

0

11

0

1

0

1

11

0

1

11

YEYEYV

dueudueudyyyyYE

dueudueudyyyyYE

uydyyduyu

dyyyyYE

yyyyydyydFyf

yy

yyF

uu

uu

Note: The EXCEL function WEIBULL(y, uses parameterization: *=

Page 35: Continuous Probability Distributions (PPT)

Weibull Density Functions (pdf)Weibull pdf's

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7 8 9 10

y

f(y)

W(1,1)W(1,2)W(2,1)W(2,2)

Page 36: Continuous Probability Distributions (PPT)

Lognormal Distribution

22

2

2

2

2222

222

**22*2

222

**

2*

log21

22

)()(

22)2(exp)2(

21)1(exp)1()(

,~)ln( :Note

otherwise 0

0,,02

1

)(

eeYEYEYV

etMeEeEYE

etMeEYE

NYY

yey

yf

YYY

YY

y

Obtaining Probabilities in EXCEL:To obtain: F(y)=P(Y≤y) Use Function: =LOGNORMDIST(y,,)

Page 37: Continuous Probability Distributions (PPT)

Lognormal pdf’sLognormal pdf's

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7 8

y

f(y)

LN(0,1)LN(0,4)LN(1,1)LN(1,4)