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RANDOM VARIABLES forUncertain Quantities
Distrete Variables Finite no. of values (e.g. binomial) Infinite no. of values (e.g. Poisson)
Continuous Variables Unbounded (e.g. normal) Bounded below (e.g. lognormal) Bounded above and below (beta)
DISCRETE PROBABILITYDISTRIBUTIONS
Finite DiscreteBinomialPoissonGeometricHypergeometric
CONTINUOUS PROBABILITYDISTRIBUTIONS
Normal Histogram beta gamma lognormal Weibull
DISTRIBUTION GALLERY from CRYSTAL BALL
DISTRIBUTION SPECIFICATIONS
DISCRETE Probability Mass Function (PMF) Cumulative Probability Function
CONTINUOUS Probability Density Function f(x) (PDF) Cumulative Distribution Function F(x)
(CDF)
Probability Mass FunctionCumulative Dist. Function
00.05
0.10.15
0.20.25
0.30.35
0.4
0 1 2 3
Prob
0
0.2
0.4
0.6
0.8
1
0 1 2 3
CumProb
FINITE DISCRETE DISTRIBUTION (example)
Value, probability pairs [ 0, 0.25] [ 1, 0.40] [ 2, 0.35]
Cumulative probability pairs [ 0, 0.25] [ 1, 0.65] [ 2, 1.00]
Mean and Variance for DISCRETE DISTRIBUTIONS
Mean = sum of pi * xi
Variance = sum of pi*xi^2 - Mean^2
St.Dev. = square root of variance
n
iiixp
1
2
1
22
n
iiixp
Binomial Distributionk = 0 to n
n = number of trialsp = success prob. On
each trialk = number of
successes in n trials0
0.1
0.2
0.3
0.4
0.5
0 1 2 3
Prob k
)!(!
!)1(),|(
knk
npppnkP knk
•n=3
•p=.4
Binomial Distribution
Mean Value or Expected Value µ = np
Variance of binomial r.v. σ² = np(1-p) or npq where q = 1-p
Standard Deviation σ = sqrt(npq) St.Dev. Ξ the square root of variance
POISSON DISTRIBUTIONk from 0 to infinity
k = number of “events” in a period of time or area of space
λ = expected number of “events” per unit time or space
P(k|λ) = (e-λ)λk/k!
00.05
0.10.15
0.20.25
0.30.35
0.4
0 1 2 3 4
Prob k
Poisson Distribution
Mean Value is the rate parameter - λ
Variance is also λ
Stdev = sqrt(λ)
CONTINUOUS DISTRIBUTION (example)
Triangular Distribution linear density drops from 2 to 0 on the unit [0,1] interval: f(x)=2 - 2x
Quadratic CDF rises from 0 to 1 on the unit interval:F(x) = 2x - x2
F(x) is the integral of f(x); f(x) is the derivative of F(x).
Mean and Variance for CONTINUOUS DISTRIBUTION
Mean = Integral of x * f(x)Variance = Integral of x2 * f(x) -
Mean^2
b
a
dxxxf )(
222 )(
b
a
dxxfx
NORMAL DISTRIBUTION
Mean is μStdev is σVariance is σ²Density function is
0
0.1
0.2
0.3
0.4
0.5
-4 -2 0 2 4
2)(
2)(5.x
exf
RULES OF THUMBbased on Normal Distribution
Pr X within 1 sigma of mean: 68.27%
Pr X within 2 sigma of mean: 95.45%
Pr X within 3 sigma of mean: 99.73%
UNIFORM DISTRIBUTION
0
0.2
0.4
0.6
0.8
1
1.2
1 2 3
UNIFORM DISTRIBUTION
Parameters: min = A, max = B
Mean Value: mean = (A + B)/2
Variance: = (B-A)2/12
HISTOGRAM DISTRIBUTION
Histogram for Reimbursed_1
0
2
4
6
8
10
12
14
<=0 0- 20 20- 40 40- 60 60- 80 80- 100 100- 120 120- 140 140- 160 >160
Category
HISTOGRAM STATISTICS
Parameters: <x0,p1,x1,p2,…,pn,xn>
Interval Midpoints mi = (xi + xi-1)/2Mean
Variance
n
iiimp
1
2
1
211
2
3
)(
n
i
iiiii
xxxxp
EXCEL FUNCTIONS FORCUMULATIVE PROBABILITY
Binomial = BINOMDIST(k,n,p,TRUE)
Poisson = POISSON(k,λ,TRUE)
Normal = NORMDIST(x,μ,σ,TRUE)