# 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 *D _{x}* 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 *f _{x}(x)* is positive. With the distributions (A-l) and (A-2), we have:

where a new variable *%* = — _{0}f integration is introduced and notation

*yj2D _{x}*

*x*

is used. The *y _{x} =* -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 = rj2D _{x}* 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 *D _{x}* 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 *Z _{corrK},_{(},_{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 = *ryj2D _{x} = l.Qj2* x 0.0400 x 1 (Г

^{4}=0.01980 kg/mm

^{3}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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