Appendix A: Convolution of Extreme Value Distribution with a Normally Distributed Variable

The objective of the analysis that follows is to obtain a convolution of the EVD

for a stationary normal random process Z(t), which is the basic distribution for this EVD (see, for example, [27]), with a stationary normally distributed random variable X(t), whose probability density distribution function is

In these formulas, the basic random process Z(t) is a homogeneous (the probability that the given level z* is exceeded depends only on the duration of the time interval and is independent of the initial moment of time) and ordinary (none of the events Z>- z” can possibly occur simultaneously with another similar event) stationary normal random process; Z ) are the extreme values of the process Z(f); N is the number of oscillations during the time interval between two adjacent upward crossings Z>z of the level z* by the process Z(t) (in such a situation the flow of the events Z >- z* is a Poisson’s process), z is the mean value of the process Z(f), x is the mean value of the process X (t) and D- and Dx are variances of the processes Z(t) and X(t). The events Z >- z* are assumed to be statistically independent. The number N in formula (A-l) is supposed to be not very small.

The probability density and the probability distribution functions of the random difference W(t) = X(t)-Z'(t) are as follows:

In these formulas, the limits of integration for the variable X (t) are defined by the range, within which the function fx(x) is positive. With the distributions (A-l) and (A-2), we have:

where a new variable % = — 0f integration is introduced and notation

yj2Dx

x

is used. The yx = -y=== ratio is the safety factor for the process X (t), and the safety

factor у. = , " for the process Z{t) can be found as y. = .' = y. = y/E. The

yJ2L)x yjlL)x

integral (A-5) determines the probability that the difference W = X{t)-Z*(t) is

below the w = rj2Dx value. When N —> F„.(w) —> 0: in a long run the process

Z*(t) will always exceed the X (i) values. When z —>", then y; —> «>, у —> and

F„,(w) —> 0: when the mean value of the process Z(t) is significant, the process Z*(t)

will always exceed the X (l) values.

When the variance ZX of the process Z(t) is significantly greater than the variance Dx of the process X(t), so that the variance ratio 8 = —X can be put equal to zero, then the integral (A-5) can be simplified: ^z

where

is the tabulated Laplace function.

Appendix B: A Numerical Integration Example

This example is given in Table B-l for the case Л = 7. The integrand is as follows:

and the corrected sum ZcorrK,(,d is computed as the sum Z minus half of the sum of the extreme ordinates.

Thus, the probability that the difference between the critical value of the SERR and its actual value will be found below the (rather high) level of the probability that the difference between the critical value of the SERR and its actual value will occur below the (rather high) level of w = ryj2Dx = l.Qj2 x 0.0400 x 1 (Г4 =0.01980 kg/mm3 is as high as 0.8384.

TABLE B.1

Numerical Integration

8.5964

8.7877

0.0001

8.8454

7.3680

0.0003

9.0964

6.0733

0.0010

9.3464

4.9036

0.0037

9.5964

3.8589

0.0115

9.8464

2.9392

0.0315

10.0964

2.1445

0.0750

10.3464

1.4748

0.1562

10.5964

0.9301

0.2849

10.8464

0.5104

0.4553

11.0964

0.2157

0.6379

11.8464

0.0460

0.8316

11.5969

0.0013

0.8464

11.8464

0.0816

0.8025

12.0964

0.2869

0.6689

12.3464

0.6172

0.4902

12.5964

1.0725

0.3161

12.8464

1.6528

0.1794

13.0964

2.3581

0.0897

13.3464

3.1884

0.0395

13.5964

4.1437

0.0153

13.8464

5.2240

0.0052

14.0964

6.4293

0.0016

14.3464

7.7596

0.0004

14.5964

9.2149

0.0001

5.9443

5.9442

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