Propagation of Uncertainty Jake Blanchard Spring 2010 Uncertainty Analysis for Engineers1

Introduction We’ve discussed single-variable probability distributions This lets us represent uncertain inputs But what of variables that depend on these inputs? How do we represent their uncertainty? Some problems can be done analytically; others can only be done numerically These slides discuss analytical approaches Uncertainty Analysis for Engineers2

Functions of 1 Random Variable Suppose we have Y=g(X) where X is a random input variable Assume the pdf of X is represented by f x. If this pdf is discrete, then we can just map pdf of X onto Y In other words X=g -1 (Y) So f y (Y)=f x [g -1 (y)] Uncertainty Analysis for Engineers3

Example Consider Y=X 2. Also, assume discrete pdf of X is as shown below When X=1, Y=1; X=2, Y=4; X=3, Y=9 Uncertainty Analysis for Engineers4

Discrete Variables Example: ◦ Manufacturer incurs warranty charges for system breakdowns ◦ Charge is C for the first breakdown, C 2 for the second failure, and C x for the x th breakdown (C>1) ◦ Time between failures is exponentially distributed (parameter ), so number of failures in period T is Poisson variate with parameter T ◦ What is distribution for warranty cost for T=1 year Uncertainty Analysis for Engineers5

Formulation 6

Plots 7 C=2 =1

CDF For Discrete Distributions If g(x) monotonically increases, then P(Y<y)=P[X<g -1 (y)] If g(x) monotonically decreases, then P(Y g -1 (y)] …and, formally, Uncertainty Analysis for Engineers8 x y x y

Another Example Suppose Y=X 2 and X is Poisson with parameter Uncertainty Analysis for Engineers9

Continuous Distributions If f x is continuous, it takes a bit more work Uncertainty Analysis for Engineers10

Example Uncertainty Analysis for Engineers11 Normal distribution Mean=0,  =1

Example X is lognormal Uncertainty Analysis for Engineers12 Normal distribution

If g -1 (y) is multi-valued… Uncertainty Analysis for Engineers13

Example (continued) Uncertainty Analysis for Engineers14 lognormal

Example Uncertainty Analysis for Engineers15

A second example Suppose we are making strips of sheet metal If there is a flaw in the sheet, we must discard some material We want an assessment of how much waste we expect Assume flaws lie in line segments (of constant length L) making an angle  with the sides of the sheet  is uniformly distributed from 0 to  Uncertainty Analysis for Engineers16

Schematic Uncertainty Analysis for Engineers17 L  w

Example (continued) Whenever a flaw is found, we must cut out a segment of width w Uncertainty Analysis for Engineers18

Example (continued) g -1 is multi-valued Uncertainty Analysis for Engineers19  <  /2  >  /2

Results Uncertainty Analysis for Engineers20 L=1 cdf pdf

Functions of Multiple Random Variables Z=g(X,Y) For discrete variables If we have the sum of random variables Z=X+Y Uncertainty Analysis for Engineers21

Example Z=X+Y Uncertainty Analysis for Engineers22

Analysis XYZPZ-rank Uncertainty Analysis for Engineers23

Result Uncertainty Analysis for Engineers24

Example Z=X+Y Uncertainty Analysis for Engineers25

Analysis XYZPZ-rank Uncertainty Analysis for Engineers26

Compiled Data zfz Uncertainty Analysis for Engineers27

Example Uncertainty Analysis for Engineers28 x and y are integers

Example (continued) Uncertainty Analysis for Engineers29 The sum of n independent Poisson processes is Poisson

Continuous Variables Uncertainty Analysis for Engineers30

Continuous Variables Uncertainty Analysis for Engineers31

Continuous Variables (cont.) Uncertainty Analysis for Engineers32

Example Uncertainty Analysis for Engineers33

In General… If Z=X+Y and X and Y are normal dist. Then Z is also normal with Uncertainty Analysis for Engineers34

Products Uncertainty Analysis for Engineers35

Example W, F, E are lognormal Uncertainty Analysis for Engineers36

Central Limit Theorem The sum of a large number of individual random components, none of which is dominant, tends to the Gaussian distribution (for large n) Uncertainty Analysis for Engineers37

Generalization More than two variables… Uncertainty Analysis for Engineers38

Moments Suppose Z=g(X 1, X 2, …,X n ) Uncertainty Analysis for Engineers39

Moments Uncertainty Analysis for Engineers40

Moments Uncertainty Analysis for Engineers41

Approximation Uncertainty Analysis for Engineers42

Approximation Uncertainty Analysis for Engineers43

Second Order Approximation Uncertainty Analysis for Engineers44

Approximation for Multiple Inputs Uncertainty Analysis for Engineers45

Example Example 4.13 Do exact and then use approximation and compare Waste Treatment Plant – C=cost, W=weight of waste, F=unit cost factor, E=efficiency coefficient Uncertainty Analysis for Engineers46 mediancov W2000 ton/y.2 F$20/ton.15 E

Solving… Uncertainty Analysis for Engineers47

Approximation Uncertainty Analysis for Engineers48

Variance Uncertainty Analysis for Engineers49