Failure probability pfPER for lognormal distribution LNjct a of E and R

1. Input parameters for E and R: jE:= 50 ctE:= 10. aE:= 0,0.1..2 x:= 0,0.1jE..3jEwE:= —

jR := 100 ctR := 10. aR :=—1,-0.9.. 2 x := 0,0.1jR.. 3jR wR :=-

Distribution parameter C given by the skewness aE:

Distribution bound x0E: x0E(aE) :=

Distribution parameter CR given by the skewness aR:

Distribution bound x0R: , (

Check:

Check:

x0R(0.000)

2. Integration bounds assuming aE>0, aR arbitrary:

x0(aE,aR) := max(x0EaE),x0R(aR)) if aR > 0 xC(0.608,0.301) = i x0E(aE) otherwise x1(aE,aR) := jR + 6-ctR if aR > 0 x(0(0.60S,—1) = ■

3. Transformation to the standardised normal distribution o(u) (for any a):

Transformed standardised variable E:

sign uE(x) otherwise aE-

otherwise aE

Standardised variable R: uR(x) = (—jR) Transformed standardised variable R:

uuR (x,aR) :=

l{

uR(x) + , . CR(aR)

) + ln(

| CR(aR) J1 + CR(aR)2)

sign (aR) ^ ln(1

+ CR(aR)2)

uR(x) otherwise

Distribution function oLN x(x) = oLN J(u) = o(uu): OR(x, aR) := pnorm(uuR(x, aR), 0, l) uuR(50,0) = -5

4. Failure probability pf using transformation to normal distribution (for aE>0, aR arbitrary:

<<E(x,aE)oR(x,aR) dx pf(0.608,0.0001) = 8.745 x 10

0.01

0.01

Figure 1. Failure probability pf=P{E>R} versus aE.

5. Alternative procedure for determination of failure probability using built-in distribution function for log-normal distributiopn o LN x(x) (for positive a only):

mE(aE) := -ln( | CE(aE) |) + ln(aE) - (0.5) • ln(1 + CE(aE)2) sE(aE) :=-J ln(1 + CE(aE)2)

Probability density of E: <)E(x,aE) := dlnorm(x- x0E«e),mE(aE),sE(aE)) <)E(50,0.0001) = 0.04

mR(aR) := -ln( CR(aR) ) + ln(aR) - (0.5) 4n(1 + CR(aR)2) sR (aR) :=-J ln(1 + CR(aR)2)

Distribution function of R: OR(x,aR) := plnorm(x - x0R(aR), mR(aR), sR (aR)) Failure probability pf (for positive a only):

Check:

<|)E(x,aE)oR(x,aR) dx pf(0.608,0.0001) = 8.745 x 10

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