Post on 30-Jan-2017
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
b
adyyf )(
1)(
dyyf
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
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 (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
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
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
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
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)
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
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
)(
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
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
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)
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)
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(
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
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
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
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)
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)
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
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
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
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)
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)
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
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
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
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
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
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)
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)
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: *=
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)
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,,)
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)