RANDOM VARIABLES for Uncertain Quantities zDistrete Variables yFinite no. of values (e.g. binomial)...

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)