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Probability Distributions for Risk Analysis
Integration TrackSCEA/ISPA Joint International Conference
Thursday, June 9th, 2011
Unit III - Module 10 1
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What This Session Is Not• What probability distributions are
– CEB 08 Probability and Statistics• Tuesday, June 26th, 1045-1215, Orange C
• How to use probability distributions in a simulation– CEA 07 Monte Carlo Simulation
• Wednesday, June 27th, 1530-1700, Orange B
• Distributions for CER risk– PT 06 CER Risk and S-Curves
• Wednesday, June 27th, 1530-1700, Orange A
• Fitting distributions to data– CEA 09 Advanced Probability and Statistics
• Thursday, June 28th, 1330-1500, Orange BUnit III - Module 10 70
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What This Session Is Not• How to do Inputs Risk
– CEB 07 Basic Cost Risk• Wednesday, June 27th, 1030-1200, Orange C
• How to analyze historical program risk data– CEA 06 Advanced Cost Risk
• Thursday, June 28th, 1030-1200, Orange B
• Joint probability distributions– INT 04 Joint Cost and Schedule Risk
• Thursday, June 28th, 1330-1500, Orange A
• Interaction of risk distributions and contract types– INT 05 Contracts Risk
• Thursday, June 28th, 1530-1700, Orange AUnit III - Module 10 71
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Outline• Probability Prerequisites
– Algebra and Calculus
• Types of Risk Distributions• Common Distributions
– Normal Distribution– Lognormal Distribution– Triangular Distribution– Other Risk Distributions
• Risk Distribution Applications• Appendix: Algebra Refresher
Unit III - Module 10 2
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Calculus for Probability• Calculus was invented (Newton, Leibniz, et al.) to support
mechanics (branch of physics)– In the modern world, and certainly for cost estimators, probability is
the primary motivation for the use of calculus
• Differential calculus deals with slopes and rates of change– Mechanics: velocity (position), acceleration (velocity)– Probability: pdf (cdf), mode
• Integral calculus deals with areas under curves and cumulative values– Mechanics: position (velocity), velocity (acceleration)– Probability: cdf (pdf), mean, variance, median, percentiles
• Derivatives and integrals are generally inverse operations– Fundamental Theorem of Calculus!
• Following slides offer a brief refresher of a few key results needed for probability derivations
Unit III - Module 10 3NEW!
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Unit III - Module 10 4
Calculus Refresher – Derivatives• Product Rule
– Derivative of a product– “Derivative of the first, times the second, plus the
first times the derivative of the second”
• Chain Rule– Derivative of a compound function– “Derivative of the outside, times derivative of the inside”
NEW!
xgxfxgxfxgdxdxfxgxf
dxdxgxf
dxd ''
xgxgfxgdxdyf
dxdyyf
dydyf
dxdxgf
dxd '''
xgy
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Unit III - Module 10 5
Calculus Refresher - Integrals• Substitution
– Technique for evaluating (definite) integrals– Replace functions, differentials, and limits consistently
– Essentially chain rule in reverse
NEW!
xgy dxxgdyxgxgdxd
dxdy ''
agFbgFyFdyyfdxxgxgf bg
ag
bg
ag
b
a '
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Unit III - Module 10 6
Calculus Refresher - Series• Taylor Series
– Representation of a function as an infinite series (sum)– Centered around a point of interest– Converges within a certain radius– Handy for computation, including approximations (finite
number of terms)– In cost estimating, tends to be used for:
• Quantities near 0, e.g., Coefficient of Variation (CV); or• Quantities near 1, e.g., Learning Curve Slope (LCS)
• Some handy Taylor Series– Log (0 ≤ x < 1)– Square root (-1 ≤ x ≤ 1)– Reciprocals (-1 < x < 1)
NEW!
xxxxx 32
1ln32
xxxxx 32
1ln32
21
8211
2 xxxx xxxxx
111
1 32
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Unit III - Module 10 7
Probability Notation• Roman letters are generally used to represent
random variables or their associated distribution functions– X, Y, Z for variables, x, y, z for realization of those variables– F, G for cumulative distribution functions (cdfs)
• f, g for corresponding probability density functions (pdfs)• Subscripts for random variable in case of ambiguity
• Lowercase Greek letters are generally used to represent parameters of probability distributions
– µ (mu) = mean, a location parameter– σ (sigma) = standard deviation, a scale parameter– α (alpha), β (beta), θ (theta), and λ (lambda) are often shape, scale, or rate
parameters
• Capital Greek letters are generally used to represent particular special functions or mathematical operations
– Γ (Gamma) = gamma function, Φ (Phi) = standard normal cdf
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Types of (Program) Risk Distributions1. Input distributions
– Characterize uncertainty of inputs to the cost estimating process, such as cost-driver parameters (weight, power, SLOC, etc.)
2. Intermediate output distributions– Characterize uncertainty about estimates for individual cost elements
• Prediction interval (PI) associated with a cost-estimating relationship (CER)– Commonly include t or log t
• Risk ranges provided by a subject matter expert (SME)
3. Final output distributions– Characterize the uncertainty of an overall cost estimate
4. Cross-program risk distributions– Characterize range of cost growth factors (CGFs) associated with historical
programs• A classic problem in risk analysis is how to make inferences about #3
from #4
Unit III - Module 10 8NEW!
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Input Distributions
1. Input distributions– Characterize uncertainty of inputs to the cost estimating process,
such as cost-driver parameters (weight, power, SLOC, etc.)– Commonly include Normal, Lognormal, and Triangular
2. Intermediate output distributions– Characterize uncertainty about estimates for individual cost elements
• Prediction interval (PI) associated with a cost-estimating relationship (CER)– Commonly include t or log t
• Risk ranges provided by a subject matter expert (SME)– Commonly include Triangular
3. Final output distributions– Characterize the uncertainty of an overall cost estimate– Commonly include Normal and Lognormal
Unit III - Module 10 9NEW!
Normality of Work Breakdown Structures, M. Dameron, J. Summerville, R. Coleman, N.St. Louis, Joint ISPA/SCEA Conference, June 2001.
Taking the Next Step: Turning OLS CER-Based Estimates into Risk Distributions. C.M. Kanick, E.R. Druker, R.L. Coleman, M.M. Cain, P.J. Braxton, SCEA 2008.
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Portfolio Risk Distributions
4. Cross-program risk distributions– Characterize range of cost growth factors (CGFs) associated with
historical programs– Commonly include Lognormal, Triangular, or other skew-right
distributions (including heavy-tailed distributions)
• A classic problem in risk analysis is how to make inferences about #3 from #4
Unit III - Module 10 10NEW!
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Unit III - Module 10 11
Continuous Distributions
• Normal and t• Lognormal• Triangular• Uniform• Other Continuous
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Unit III - Module 10 12
Normal Distribution Overview• Distribution • Parameters and Statistics
– Mean = – Variance = 2
– Skewness = 0– 2.5th percentile = - 1.96– 97.5th percentile = + 1.96
• Key Facts– If X ~ N(, ), then ~ N(0,1)
(“standard” normal)– Central Limit Theorem holds for n ≥ 30 – 68.3/95.5/99.7 Rule– Limiting case of t distribution– Exponential of normal is lognormal– Dist: NORMDIST(x, mean, stddev, cum)
• cum = TRUE for cdf and FALSE for pdf– Inv cdf: NORMINV(prob, mean, stddev)
• Applications– Central Limit Theorem
• Approximation of distributions
– Regression Analysis• Assumed error term
– Distribution of cost• Default distribution
– Distribution of risk• Symmetric risks and uncertainties
)( X
Normal Distribution
0.0
0.1
0.2
0.3
0.4
-4 -3 -2 -1 0 1 2 3 4
2
2
2
21
x
exp
x
t
dtexxF 2
2
2
21
),( x
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Unit III - Module 10 13
t Distribution Overview• Distribution • Parameters and Statistics
– Degrees of freedom = n– Mean = 0– Variance = – Skewness = 0
• Key Facts– As n approaches infinity, the t
distribution approaches Normal– The t is distributed as – Excel
• cdf = TDIST(x, n, tails)– Tails = 1 or 2 depending on
whether you want to include the probability in the left-hand tail
• Inv cdf = TINV(prob, n)
• Applications– Confidence Intervals
• Mean of Normal variates
– Regression Analysis• Significance of individual
coefficients
– Hypothesis Testing
2/)1(2 )]/(1)[2/(
]2/)1[(
nnxnnnxp
2n
n),( x
nnNt
/)()1,0(
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
-4 -3 -2 -1 0 1 2 3 4
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Unit III - Module 10 14
Lognormal Distribution Overview• Distribution • Parameters and Statistics
– Median = – Std Deviation of ln X = σ– Mean = – Variance =
• Key Facts– If X has a lognormal distribution, then ln(X)
has a normal distribution– For small standard deviations, the normal
approximates the lognormal distribution• For CVs < 25%, this holds
– Excel• Cdf = LOGNORMDIST(x, mean, stddev)• Inv cdf = LOGINV(prob, mean, stddev)
• Applications– Risk Analysis
2
2
2)/ln(exp
21
x
xxp
22e
1222 ee
Lognormal Distribution
0
0.01
0.02
0.03
0.04
0.05
0.06
0 5 10 15 20 25 30 35
),0[ x
10
e
NEW!
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Unit III - Module 10 15
Triangular Distribution Overview• Distribution • Parameters and Statistics
– Min = a– Max = b– Mode = c (a ≤ c ≤ b)– Mean =
– Variance =
• Key Facts– Excel
• pdf and cdf calculations can be handled as they are listed in the above formulas
– A symmetrical triangle approximates a normal when
• Applications– Risk Analysis
• SME Input
bxccbabxbcxaacabax
xp/2/2
bxc
cbabxb
cxaacab
ax
xF 2
2
1
6,,6 bca
3cba
18
222 bcacabcba
0.0000.0050.0100.0150.0200.0250.0300.035
0 20 40 60 80 100 120
19
NEW!
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Unit III - Module 10 16
Uniform Distribution Overview• Distribution • Parameters and Statistics
– Min = a– Max = b– Mode = any value [a,b]– Mean =
– Variance =
• Key Facts– Excel
• pdf and cdf calculations can be handled as they are listed in the above formulas
• Applications– Risk Analysis
• SME Input– Sampling from arbitrary
distributions• Example: Rejection Sampling
2ba
12
2ab
19
‐0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.00 5.00 10.00 15.00 20.00
Uniform Distribution
bxorax
bxaabxp
0
1
bx
bxaabax
axxF
1
0
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Unit III - Module 10 17
Beta Distribution Overview• Distribution • Parameters and Statistics
– Min = 0– Max = 1– Mode =
– Mean =
– Variance =
• Key Facts– Excel
• pdf and cdf calculations can be handled as they are listed in the above formulas
• Applications– Risk Analysis– Order Statistics– Rule of Succession– Bayesian Inference– Task Duration
11 1
xxxf ,xIxF
12
19
0
0.5
1
1.5
2
0.00 0.20 0.40 0.60 0.80 1.00 1.20
Beta Distribution
1 ,1for 2
1
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Unit III - Module 10 18
Gamma Distribution Overview• Distribution • Parameters and Statistics
– Min = 0– Max = ∞– Mode =
– Mean =
– Variance =
• Key Facts– Excel
• GAMMADIST, GAMMAINV, and GAMMALN [natural log of gamma]
– It is the conjugate prior for the precision (i.e. inverse of the variance) of a normal distribution.
• Applications– Risk Analysis
– Due to shape and scale parameters
– Modeling– Size of insurance claims
and rainfall
xexxp
1
xxF
,1
2
19
0
0.01
0.02
0.03
0.04
0.05
0.06
0.00 20.00 40.00 60.00 80.00 100.00 120.00
Gamma Distribution
1for 1
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Unit III - Module 10 19
Weibull Distribution Overview• Distribution • Parameters and Statistics
– Min = 0– Max = ∞– Mode =
– Mean = – Variance =
• Key Facts– Excel
• WEIBULL function where α = k and β = λ, as shown above
– Interpolates between the exponential distribution with intensity 1/λ when k =1 and a Rayleigh Distribution of when k = 2.
• Applications– Risk Analysis– Reliability Engineering– Failure Analysis– Delivery Times– RF Dispersion
kxexF
1
k11 22 21 k
19
‐0.1
0
0.1
0.2
0.3
0.4
0.5
0.00 2.00 4.00 6.00 8.00 10.00 12.00
Weibull Distribution
00
01
x
xexkxf
kxk
1 0
1 11
k
kk
k k
NEW!
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Unit III - Module 10 20
Exponential Distribution Overview• Distribution • Parameters and Statistics
– Min = 0– Max = ∞– Mode = 0– Mean =
– Variance =
• Key Facts– Excel
• EXPONDIST– Exponential distribution exhibits
infinite divisibility– Memoryless
• Applications– Risk Analysis– Inter-arrival times (Poisson)– Time between events– Reliability engineering– Hazard Rate– Bathtub Curve
.0,0,0
xxe
xfx
.0,0,0,1
xxe
xFx
1
2
19
0
0.05
0.1
0.15
0.2
0.00 10.00 20.00 30.00 40.00 50.00 60.00
Exponential Distribution
0, allfor PrPr tstTsTtsT
NEW!
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0
0.05
0.1
0.15
0.00 20.00 40.00 60.00 80.00 100.00 120.00
Pareto Distribution
Unit III - Module 10 21
Pareto Distribution Overview• Distribution • Parameters and Statistics
– Min =– Max = +∞– Mode =– Mean =
– Variance =
• Key Facts– Excel
• pdf and cdf calculations can be handled as they are listed in the above formulas
– The Pareto distribution and log-normal distribution are alternative distributions for describing the same types of quantities.
• Applications– Risk Analysis– Population sizes– File size and Internet traffic– Hard disk error rates– Distribution of Income
m
mm
xx
xxxx
xffor 0
for 1
m
mm
xx
xxxx
xFfor 0
for 1
mm xx
xx
for 1
2for
21 2
2
mx
19mx
mx
NEW!
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Unit III - Module 10 22
Discrete Distributions
• Bernoulli• Binomial• Other Discrete
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Unit III - Module 10 23
Bernoulli Distribution Overview• Distribution • Parameters and Statistics
– Min = 0– Max = 1– Mean = p– Variance = pq
• Key Facts– Excel
• pdf and cdf calculations can be handled as they are listed in the above formulas
– The sum of n Bernoullis is Binomial (n,p)
• Applications– Risk Analysis
• Discrete Risks• X = Cf * Bernoulli• p = Pf
101
xpxpq
xp
1110
00
xxq
xxF
NEW!
q p
0 1
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Unit III - Module 10 24
Binomial Distribution Overview• Distribution • Parameters and Statistics
– Min = 0– Max = n– Mean = np– Variance = np(1-p)
• Key Facts– Excel
• BINOMDIST– The number of “successes” in a
sequence of n independent experiments
– Beta Distribution is the conjugate prior
• Applications– Risk Analysis– Models for repeated processes
or experiments– May be used for batch
processing failures
knk ppkn
kp
1
x
i
ini ppin
xF0
1
0
0.02
0.04
0.060.08
0.1
0.120.14
0.00 10.00 20.00 30.00 40.00 50.00 60.00
Binomial Distribution
NEW!
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Unit III - Module 10 25
Negative Binomial Distribution Overview
• Distribution • Parameters and Statistics– Min = 0– Max = ∞– Mean =
– Variance =
• Key Facts– Excel
• NEGBINOMDIST
• Applications– Risk Analysis
• Discrete Risks– Process Analysis
kr ppkrk
kp
1
1 rkIkX p ,11Pr
0
0.02
0.04
0.06
0.08
0.1
0.00 10.00 20.00 30.00 40.00 50.00 60.00
Negative binomial Distribution
ppr1
21 ppr
NEW!
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Unit III - Module 10 26
Poisson Distribution Overview• Distribution • Parameters and Statistics
– Min = 0– Max = ∞– Mean = λ– Variance = λ
• Key Facts– Excel
• POISSON– Ladislaus Bortkiewicz, 1898, used
to investigate the number of Prussian soldiers killed by horse kick.
– Law of Small Numbers or Law of rare events
• Applications– Risk Analysis– Reliability Engineering– Customer Service– Civil Engineering– Astrology
!k
expk
k
i
i
iexF
0 !
0
0.02
0.04
0.06
0.08
0.1
0.12
0.00 10.00 20.00 30.00 40.00 50.00 60.00
Poisson Distribution
NEW!
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Unit III - Module 10 27
Geometric Distribution Overview• Distribution • Parameters and Statistics
– Min = 1– Max = ∞– Mean = 1/p– Variance =
• Key Facts– Excel
• pdf and cdf calculations can be handled as they are listed in the above formulas
– Continuous analogue is the exponential distribution
• Applications– Risk Analysis
• Discrete Risks
ppxp k 11 kpxF 11
0
0.05
0.1
0.15
0.2
0.00 10.00 20.00 30.00 40.00 50.00 60.00
Geometric Distribution
2
1p
p
NEW!
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Unit III - Module 10 28
Discrete Uniform Distribution Overview
• Distribution • Parameters and Statistics– Min = a– Max = b– Mean =
– Variance =
• Key Facts– Excel
• pdf and cdf calculations can be handled as they are listed in the above formulas
– German Tank Problem
• Applications– Risk Analysis
• Discrete Risks• Quantity selections
Otherwise0
11 bkaabxp
bk
bkaabak
akxF
111
0
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.00 10.00 20.00 30.00 40.00 50.00 60.00
Discrete Uniform Distribution
2ba
12
11 2 ab
NEW!
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Unit III - Module 10 29
Befriending Your Distributions
• Normal Demonstration• Triangular Derivations• Lognormal Derivations
and Demonstration
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Befriending Your Distributions• Probability distributions are to risk analysts
what words are to poets• Look at distributions in multiple ways:
– Graphical – PDF and CDF graphs– Numerical – Excel functions– Algebraic – formulae, parameters
• Build a “toy problem” and play with it• Understand properties of distributions, when
they “arise,” how they are related to other distributions
Unit III - Module 10 30NEW!
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Excel 2010 Distributions• New and improved format for statistical functions in Excel
– More consistent syntax– Excel 2007 versions maintained for backwards compatibility
• Same set of distributions– Continuous: BETA, CHISQ, EXPON, F, GAMMA, GAMMALN, LOGNORM,
NORM, T, WEIBULL– Discrete: BINOM, HYPGEOM, NEGBINOM, POISSON
• Suffixes denote different variants– .DIST = cdf (and sometimes pdf)– .S = standard (normal only)– .INV = inverse cdf– .2T = two-tail– .RT = right tail (left tail is default)
• Examples:– =NORM.S.INV(RAND()) will generate a standard normal– =T.INV.2T(0.05,30) will give the (positive) critical value for a t-test at alpha =
0.05, 30 degrees of freedom
Unit III - Module 10 31NEW!
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Triangular Distribution –PDF and Mean
• For Triangle(L,M,H) , denote L=a, H=b, M=c by T(a,c,b)• Since the area of the triangle must be 1 (100%), the height is
twice the reciprocal of the base– We can then derive the PDF by using similar triangles
bxc
cbxb
ab
cxaacax
abxp 2
2
dxcbxb
abxdx
acax
abxdxxxpXE
b
c
c
a
b
a
22
b
c
c
a
cb
xbx
ac
axx
ab
3223
32
32
1
222222
32
32
32
32
32
321 cbcbbcbaacaacc
ab
3331 22 cbaabacbc
ab
NEW!Unit III - Module 10
“Understatement of Risk and Uncertainty by Subject Matter Experts (SMEs)”, P.J. Braxton, R.L. Coleman, SCEA/ISPA, 2011.
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Unit III - Module 10
Triangular Distribution – Variance
NEW!
“Understatement of Risk and Uncertainty by Subject Matter Experts (SMEs)”, P.J. Braxton, R.L. Coleman, SCEA/ISPA, 2011.
1818
218
2222222 cbacabbcacabcaabbbcacabcba
2222 XEXE
b
c
c
a
cb
xbx
ac
axx
ab
4334
21
32
32
21
1
32232233223223
21
32
32
211 cbccbbbccbbacaacacaacc
ab
dxcbxb
abxdx
acax
abxdxxpxXE
b
c
c
a
b
a
2222 22
62
132 222
222222 bcacabcbaaabbacbccaabbacbcc
9222
3
22222 bcacabcbacba
18
44422218
333333 22222222 bcacabcbabcacabcbaXE
Square of the base minus product of the
half-bases!
1818
22 2222222 cbaccbacaccabbccbcbaacc
Sum of squares of half-bases and product of half-bases!
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Substituting a Triangular for a Normal:The √6 Factor
• For a symmetric triangle, let ML = m, L = m-w, H = m+w, where w is the half-base– Then the mean is m, and the variance is w2/6
• It follows that the half-base is greater than the standard deviation by a factor of √6
• To approximate a normal, N(μ, σ) the factor of √6 is multiplied by the standard deviation of the normal to be emulated to produce the half-base– By this means, end points are found
that will produce a triangular distributionthat emulates the underlying normal inmean and standard deviation
– This triangular distribution, Tri(μ -√6σ, μ, μ +√6σ)differs from the underlying normal in all othermoments, and at all percentiles other than themedian and two “cross-over” points, but thedifference is minor 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
‐3.0 ‐2.5 ‐2.0 ‐1.5 ‐1.0 ‐0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0
Tri
Norm
NEW! 34Unit III - Module 10
“Understatement of Risk and Uncertainty by Subject Matter Experts (SMEs)”, P.J. Braxton, R.L. Coleman, SCEA/ISPA, 2011.
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Unit III - Module 10 35
Derivation of Lognormal pdf• Relationship of Lognormal (X) and
corresponding Normal (Y)
• Lognormal cdf in terms of related Normal cdf
• Lognormal pdf is derivative of cdf
– Apply chain rule
NEW!
,~ln,~ NXYLogNeX Y
xFxYPxXPxF YX lnln
2
2
2ln
21ln1lnln
x
YY ex
xfx
xdxdxF
dyd
Both parametrized in terms of mean and
standard deviation of the related normal
xFdxdxF
dxdxFxf YXXX ln'
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Unit III - Module 10 36
Derivation of Lognormal Median• Definition of median
• Lognormal pdf (see previous slide)
• Substitution
• This is just the pdf of the related normal!
• Median is preserved by transformation
NEW!
21
21
02
ln2
2
mx
dxex
dxx
dyxy 1ln
emm ln5.0
21
0
m
X dxxf
21
21ln
2 2
2
m
y
dye
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• Definition of mode, apply product rule and chain rule!
• First term is strictly positive, get common denominator
• pdf is concave down at peak– We’ll spare you the derivative!
22
222
2
11
ln0ln 2
CVeCVeexx
xx
Unit III - Module 10 37
Derivation of Lognormal Mode
NEW!
0ln12
12
1' 2222
ln2
ln2
2
2
2
xx
xee
xdxdxf
xx
X
Note mode shift (left)
See later CV derivation
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Unit III - Module 10 38
Derivation of Lognormal Mean• Definition of expected value, lognormal pdf
(see earlier slide), substitution
• Completing the square!
• Normal pdf, area under the curve = 1!
NEW!
dyeedxedxxxfXE yyx
X2
2
2
2
20
2ln
0 21
21
dyedyeyyyy
2
4222
2
222
22
222
21
21
dyedxex yy
211
21 2
2222
2
2
222
CVeCVeedyeey
Note mean shift (right)
See later CV derivation
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Unit III - Module 10 39
Derivation of Lognormal Variance• Same approach, substitution
• Recall “easy-to-compute” variance
NEW!
dyeedxexdxxfxXE yyx
X22
02
ln
0
22 2
2
2
2
21
2
dyedyeyyyy
2
4222
2
222
2442
242
21
21
dyedxex yy
22
22
2 2222
22
21
edyee
y
12222 222222 eeeeXEXEXVar
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Unit III - Module 10 40
Derivation of Lognormal CV• Standard deviation is square root of variance
• First factor is the mean!
• Note that the CV is entirely a function of the variance (not CV) of the related normal– Pythagorean relationship– Normal standard deviation as function of lognormal CV
NEW!
12
2
2
ee
12
eCV
221 eCV
CVCVCV 222 1ln1ln
std dev (norm)
CV (lognorm)
0.100 10%0.198 20%0.294 30%0.385 40%0.472 50%
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2
2
2
2
2
2
2
1ln21ln
1ln2
2
XEXVarXE
XEXVar
XEeee
XEXVar
XEXE
XE
Unit III - Module 10 41
Derivation of Related Normal• Manipulate first and second moment to solve
for normal parameters• Variance of related normal as function of
variance and mean of lognormal
• Mean of related normal as function of variance and mean of lognormal– Divide top and bottom by mean of lognormal
NEW!
22
2
22
2
2
2
2
1ln2
2
2
XEXVare
ee
XEXEXVar
XEXE
Note mean shift (left)
Agrees with previous CV result
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CV (lognorm)
mode shift factor
mean shift factor
percentile of mean
10% 0.990 1.005 52.0%20% 0.962 1.020 53.9%30% 0.917 1.044 55.8%40% 0.862 1.077 57.6%50% 0.800 1.118 59.3%
Related Normal Example• Mean = 100, CV = 20%
– Mode shift = -3.8% (relative to median)– Mean shift = +2.0% (relative to median)
Unit III - Module 10 42
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
0.022
0.024
0.026
0.028
0.03
‐50 0 50 100 150 200 250 300
Lognormal
Normal
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
0 1 2 3 4 5 6 7 8 9 10
Related Normal Check
Related Normal
Mean = 100.0
Median = 98.0Mode = 94.3Mean = ln(98.0) = 4.6
Std dev = 0.2
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Unit III - Module 10 43
Selecting From a Distribution
• Random Number Generation
• Inverse CDF Technique
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Unit III - Module 10 44
The Inverse CDF Technique• Monte Carlo Simulation requires a means to generate probability
distributions– All events in a simulation must be assigned a value from some sort of
probability distribution– The Inverse CDF Technique is the easiest and most common
• The CDF of a distribution maps a value (x) to the probability that the random variables takes on a value less an or equal to x– Therefore, the inverse of the CDF maps a probability (between 0 and 1) to
a value from the distribution– By generating a random uniform(0,1) random number, we can produce a
value from any distribution with an invertible CDF• Every simulation must contain some sort of Uniform (0,1) random
number generator– In Excel this function is “=RAND()”, although in terms of “randomness” it is
not sufficient– There is an entire area of computer science dedicated to the production of
the “most random” numbers (of particular interest to cryptographers)
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Unit III - Module 10 45
Inverse CDF Normal Example• The RAND() function
returns a value of 0.0639• This is plugged into the
inverse CDF• The value -45.677 is
returned– 1.523 standard deviations
below the mean
• This is at the appropriate percentile of the distribution
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-90 -60 -30 0 30 60 90
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
-90 -60 -30 0 30 60 90
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Unit III - Module 10 46
Applications of the Triangular Distribution
• Subject Matter Expert (SME) Understatement of Uncertainty
• Conflating SME Inputs
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The Geometry of Symmetric Triangles• For a symmetric Triangle(L, M, H), where M-L = H-M• Find points l and h such that l and h are the pth and 1-pth percentiles
If l-L = 1/2*(M-L), H-h = 1/2*(H-M), then p = 1/(2*22) = 1/8 = 12.5%If l-L = 1/3*(M-L), H-h = 1/3*(H-M), then p = 1/(2*32) = 1/18 = 5.6%pth percentile → √(p/2) base fraction → √(2p) half-base fraction So, the 20th percentile -> 1/5 occurs at point √(1/10) = 0.3162 base fraction
L l M h H
These two “tiled pictures” show two relationships of a fraction of
the base to a fraction of the area, showing the above
equations in a graphical way.
13
1
9
12
1
4
NEW!Unit III - Module 10
“Understatement of Risk and Uncertainty by Subject Matter Experts (SMEs)”, P.J. Braxton, R.L. Coleman, SCEA/ISPA, 2011.
L l M h H
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Triangles With Related Areas• We wish to know how to draw triangular distributions that are related
to one another• Constant area:
– Used in expansion of experts (correcting understated variance)– For area to remain constant, in this case A = 1, as the base increases by
a factor, the height must be multiplied by the reciprocal of that factor
• Reduction in area:– For area to be reduced by a factor, the dimensions of a similar triangle
must be reduced by the square root of that factor
– For area to be reduced by a factor, the height must be reduced by that factor if the base is to remain constant
• Used in sampling of experts
khbkbhA
21
21
k
hk
bhbk
Ak
A 111112 2
1211
khbhb
kA
kA 1
11112 21
211
NEW!Unit III - Module 10
“Understatement of Risk and Uncertainty by Subject Matter Experts (SMEs)”, P.J. Braxton, R.L. Coleman, SCEA/ISPA, 2011.
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Correction of Understated Variance for Triangles
• For symmetric triangles– To expand from 20-80 to Min-Max,
multiply by 2.72 = 1/0.368– √(1/10) = 0.3162 base fraction– √(2/5) = 0.6325 half-base fraction
– To expand from plus-or-minus-one-sigma to Min-Max, multiply by 2.45 (√6)– (√6-1)/2√6 = 0.2959 base fraction– (√6-1)/√6 = 0.5918 half-base fraction– Compare with 68.3% within
one sigma rule of thumb forNormal distribution
20 2060
0.6320.368
0.684 0.316
17.5 17.565
0.5920.418
0.704 0.296
NEW!Unit III - Module 10
“Understatement of Risk and Uncertainty by Subject Matter Experts (SMEs)”, P.J. Braxton, R.L. Coleman, SCEA/ISPA, 2011.
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Correction of Understated Variance for Triangles
• For symmetric triangles– General case– To expand from pth-(1-p)th to Min-Max, multiply by 1/(1-√(2p))– If 2p = α, then multiply by 1/(1-√ α)– To expand from (α1/2)th-(1-α1/2) th to
(α2/2)th-(1-α2/2) th [α1 > α2],multiply by (1-√ α 2)/(1-√ α 1)
– For example, to expand from33-67 to 20-80, multiply by(1-√ (2/5)/(1-√ (2/3)) ≈ 2.0
p p1-2p
√(2p)1-√(2p)
1-√(p/2) √(p/2)
“Understatement of Risk and Uncertainty by Subject Matter Experts (SMEs)”, P.J. Braxton, R.L. Coleman, SCEA/ISPA, 2011.
50Unit III - Module 10 NEW!
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Variance of Hybrid Distributions –A Pythagorean Relationship
• Suppose k distributions with pdf pi(xi), mean μi, and standard deviation σi are sampled
• Then the pdf of the hybrid distribution is the “average” of the pdfs
• The mean of the hybrid distribution is the average of the means
• The variance of the hybrid distribution is the average of the variances plus the variance of the means taken as a discrete probability distribution!
– See next slide for derivation
k
iii xp
kxp
1
1
k
dxxpxk
XE
k
iik
iiiii
1
1
1
NEW!Unit III - Module 10
“Understatement of Risk and Uncertainty by Subject Matter Experts (SMEs)”, P.J. Braxton, R.L. Coleman, SCEA/ISPA, 2011.
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Unit III - Module 10
Variance of Hybrid Distributions –A Pythagorean Relationship
• In the special case of two congruent distributions with centers at m-d and m+d, the variance is
kdxxpx
kXE
k
iiik
iiiii
1
22
1
22 1
2
1
1
22
222 1
k
ii
k
iii
kkXE
2
1
1
2
1
2
1 k
ii
k
ii
k
ii
kkk
22222
2
2dmdmdm
NEW!
“Understatement of Risk and Uncertainty by Subject Matter Experts (SMEs)”, P.J. Braxton, R.L. Coleman, SCEA/ISPA, 2011.
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Equivalence of Averaging Distributions and Averaging Parameters for Symmetric Triangles• In the case of symmetric triangles, averaging the individual triangles (with
perfect rank correlation) can be shown to be equivalent to averaging the parameters
– We will prove it in the case of two triangles, but the proof can easily be extended to more
• As previously shown, the pth percentile (p<0.5) for a symmetric triangle is at the √(2p) half-base fraction
– So the pth percentiles of the two triangles and their average are:
– But this is clearly just the pth percentile of the average distribution
– A similar proof works for p>0.5– Since all percentiles are equal, the resulting distributions are identical
• Monte Carlo simulation could be used to explore the difference between the two methods for asymmetric triangles, but it is not expected to be large
2
22
22 221121222111
acacpaaacpaacpa
222
2212121 aaccpaa
NEW!Unit III - Module 10
“Understatement of Risk and Uncertainty by Subject Matter Experts (SMEs)”, P.J. Braxton, R.L. Coleman, SCEA/ISPA, 2011.
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Equivalence of Averaging Means andAveraging Modes for Triangles
• If we average parameters, as long as we average mins and maxes, it doesn’t matter whether we average means or modes
– Algebraically equivalent– Any number of triangles, symmetry not required
• Let the kth triangle be T(ai, ci, bi), and parameter-averaged triangle beT(A, C, B), where
• This is averaging the modes; the resulting mean is
which is just the average of the means!• Reversing the flow, averaging the means can be shown to produce a mode
which is the average of the modes
k
bB
k
cC
k
aA
k
ii
k
ii
k
ii
111
k
cba
k
cbaCBA
k
i
iiik
ii
k
ii
k
ii
1111 3
33
NEW!Unit III - Module 10
“Understatement of Risk and Uncertainty by Subject Matter Experts (SMEs)”, P.J. Braxton, R.L. Coleman, SCEA/ISPA, 2011.
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Unit III - Module 10 55
Cost Growth Factor and Percentiles
• A result for Final Output Distributions
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Implied Percentile or CGF• For a given CV, Cost Growth Factor (CGF) and
percentile (of the point estimate) are related– Think of CGF as the factor it take to bring the point estimate
up to the true mean– Different answers for normal of lognormal
• Given a CGF and CV, we might ask:– At what percentile must the point estimate be?
• Given a percentile and CV, we might ask:– What is the CGF between that value and the mean?
Unit III - Module 10 56NEW!
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Implied Percentile – Normal• Without loss of generality (w.l.o.g.), assume mean of
one (1.0)– Then the standard deviation is equal to the CV!
• We are looking for the percentile of the reciprocal of CGF
Unit III - Module 10 57
1,0~,~ NXZNX
CV
CGFCGFp 1/1/1
NEW!
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Implied Percentile – Lognormal• W.l.o.g., assume lognormal mean of one (1.0)• Then the mean and standard deviation of the related
normal are:
• We are looking for the percentile of the reciprocal of CGF (unit space)
Unit III - Module 10 58
211lnCV
21ln CV
1,0~,~ln,~ NYZNXYLogNeX Y
2
2
1ln
1ln/1ln
CV
CGFCV
CGFp
NEW!
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Percentile-to-CGF Conversion• This table shows the implied CGF for a given percentile of
the Point Estimate for various CVs of a Normal distribution
59
NORMAL
p 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.500.05 1.09 1.20 1.33 1.49 1.49 1.49 2.36 2.92 3.85 5.630.10 1.07 1.15 1.24 1.34 1.34 1.34 1.81 2.05 2.36 2.780.15 1.05 1.12 1.18 1.26 1.26 1.26 1.57 1.71 1.87 2.080.20 1.04 1.09 1.14 1.20 1.20 1.20 1.42 1.51 1.61 1.730.25 1.03 1.07 1.11 1.16 1.16 1.16 1.31 1.37 1.44 1.510.30 1.03 1.06 1.09 1.12 1.12 1.12 1.22 1.27 1.31 1.360.35 1.02 1.04 1.06 1.08 1.08 1.08 1.16 1.18 1.21 1.240.40 1.01 1.03 1.04 1.05 1.05 1.05 1.10 1.11 1.13 1.150.45 1.01 1.01 1.02 1.03 1.03 1.03 1.05 1.05 1.06 1.070.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.000.55 0.99 0.99 0.98 0.98 0.98 0.98 0.96 0.95 0.95 0.940.60 0.99 0.98 0.96 0.95 0.95 0.95 0.92 0.91 0.90 0.890.65 0.98 0.96 0.95 0.93 0.93 0.93 0.88 0.87 0.85 0.840.70 0.97 0.95 0.93 0.91 0.91 0.91 0.84 0.83 0.81 0.790.75 0.97 0.94 0.91 0.88 0.88 0.88 0.81 0.79 0.77 0.750.80 0.96 0.92 0.89 0.86 0.86 0.86 0.77 0.75 0.73 0.700.85 0.95 0.91 0.87 0.83 0.83 0.83 0.73 0.71 0.68 0.660.90 0.94 0.89 0.84 0.80 0.80 0.80 0.69 0.66 0.63 0.610.95 0.92 0.86 0.80 0.75 0.75 0.75 0.63 0.60 0.57 0.55
Coefficient of Variation (CV)
Unit III - Module 10
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LOGNORMAL
p 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.500.05 1.09 1.18 1.29 1.41 1.55 1.69 1.85 2.03 2.22 2.430.10 1.07 1.14 1.22 1.31 1.41 1.52 1.64 1.76 1.90 2.050.15 1.05 1.11 1.18 1.25 1.33 1.42 1.51 1.61 1.71 1.820.20 1.04 1.09 1.15 1.20 1.27 1.34 1.41 1.49 1.57 1.660.25 1.04 1.07 1.12 1.17 1.22 1.27 1.33 1.40 1.46 1.540.30 1.03 1.06 1.09 1.13 1.17 1.22 1.27 1.32 1.37 1.430.35 1.02 1.04 1.07 1.10 1.13 1.17 1.21 1.25 1.29 1.340.40 1.01 1.03 1.05 1.07 1.10 1.12 1.15 1.19 1.22 1.260.45 1.01 1.02 1.03 1.05 1.06 1.08 1.11 1.13 1.16 1.190.50 1.00 1.00 1.01 1.02 1.03 1.04 1.06 1.08 1.10 1.120.55 0.99 0.99 0.99 0.99 1.00 1.01 1.02 1.03 1.04 1.050.60 0.99 0.98 0.97 0.97 0.97 0.97 0.97 0.98 0.98 0.990.65 0.98 0.97 0.95 0.94 0.94 0.93 0.93 0.93 0.93 0.930.70 0.98 0.95 0.94 0.92 0.91 0.90 0.89 0.88 0.88 0.870.75 0.97 0.94 0.91 0.89 0.87 0.86 0.84 0.83 0.82 0.810.80 0.96 0.92 0.89 0.86 0.84 0.82 0.80 0.78 0.76 0.750.85 0.95 0.91 0.87 0.83 0.80 0.77 0.74 0.72 0.70 0.690.90 0.94 0.88 0.84 0.79 0.75 0.72 0.69 0.66 0.63 0.610.95 0.92 0.85 0.79 0.74 0.69 0.64 0.61 0.57 0.54 0.51
Coefficient of Variation (CV)
Percentile-to-CGF Conversion• This table shows the implied CGF for a given percentile of
the Point Estimate for various CVs of a Lognormaldistribution
60Unit III - Module 10
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Unit III - Module 10 61
Alternate Specifications
• Normal• Lognormal• Two Lognormals:
Ambiguity from Square Roots
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Alternate Specification of NormalGiven Mean Std Dev CV
Mean,CV
Mean,Percentile (Xp, p)
CV,Percentile (Xp, p)
Two Percentiles
Unit III - Module 10 62
pZ p1
CV CV
p
p
ZX
p
p
Z
X
1
CVZX
p
p
1 CVZCVX
p
p
1CV
2,1
1
ipZ ii
12
2112
ZZXZXZ
12
12
ZZXX
NEW!
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Alternate Specification of LognormalGiven Mean Std Dev CV µ σ
Mean,CV
Mean,Percentile (Xp, p)
CV,Percentile (Xp, p)
TwoPer-centiles
Unit III - Module 10 63
XE XECV CV
12
e
XEX
Z
Z
pp
p
ln22
21ln
CVXE 21ln CV
XE
pZ p1
21ln
CVXE
CV 21lnln
CVZX
p
p
2
2e
XECV
XECV
2,1
1
ipZ ii
2
2e XECV 1
2
e12
2112 lnlnZZ
XZXZ
12
1
2ln
ZZXX
21ln CV
NEW!
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Alternate Specification of LognormalGiven Mean Std Dev CV µ σ
Mean, Median (m)
Median(m), Mode (M)
Mean, Mode (M)
Unit III - Module 10 64
XE XECV
12
e
mXEln2 mln
XE MXE 2ln31
2
2e XECV
XECV
12
e
12
e
mln
Mmln
MXEln
32
NEW!
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Alternate Specification of LognormalGiven Mean Std Dev CV µ σ
Median (m),Percentile (Xp, p)
Mode (M),Percentile (Xp, p)
Unit III - Module 10 65
12
e
MX
Z
Z
pp
p
ln421
22
p
p
ZmX
ln
pZ p1
21ln CVM XECV 2
2e
12
e XECV 2
2e mln
NEW!
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Two Lognormals• The plus-or-minus in the two previous orange cells gives rise to the
possibility of two solutions to a given specification• Mean and a Percentile
– Percentile > Mean: two tenable solutions– Percentile = Mean: one tenable solution
and one singular solution (σ = 0)– Percentile < Mean: one tenable solution
and one untenable solution (σ < 0)
• Mean and a Percentile– Percentile < Mode: two tenable solutions– Percentile = Mode: one tenable solution
and one singular solution (σ = 0)– Percentile > Mode: one tenable solution
and one untenable solution (σ < 0)
• Note that the mischief occurs when the percentile (load) is on the other side of the mean/mode (effort) from the median (fulcrum)
Unit III - Module 10 66
XEX
ZZ ppp ln22
MX
ZZ p
pp ln4
21
22
from the quadratic formula!
third-class lever!
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0
0.005
0.01
0.015
0.02
0.025
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100 120 140 160
Two Lognormals Example• Mean = 100, 80th percentile = 120
– “Regular” solution with 26% CV– “Extreme” solution with 258% CV!
Unit III - Module 10 67Mean = 100.0
Median = 96.8
80th percentile = 120Mode = 4.7
Median = 36.1
Mode = 90.7
NEW!