By Vladimir Troyan, Yurii Kiselev
This textbook features a attention of the vast box of difficulties hooked up with statistical equipment of processing of saw information, with the most examples and thought of versions regarding geophysics and seismic exploration. This textbook may be really useful to scholars and pros from a variety of fields of physics, attached with an estimation of the parameters of the actual items by means of experimental information. The reader may also locate many vital themes, that are the root for statistical tools of estimation and inverse challenge ideas.
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Extra resources for Statistical Methods of Geophysical Data Processing
3 Semi-invariants or cumulants If for one-dimensional distribution the moments of s-th order exist, then semiinvariants (cumulants) exist also, which are defined as coefficients κ1 , κ2 , . . , κs of expansion of the logarithm of the moment generating function in Taylor series ln αξ (t) = κ1 t + κ2 t3 t2 + κ3 + . . 2! 3! 4 , t=0 3µ22 and so on. Exercises (1) To find the characteristic function gξ (t) of the random variable ξ with a probability density 1 fξ (x) = e−|x| . 2 (2) The random variable ξ has the characteristic function 1 .
What will happen n to their arithmetic average η = ξi /n under increasing of n? i=1 (4) The random variables ξ1 , ξ2 , . . , ξn are distributed uniformly on the intervals n (−1, 1), (−2, 2), . . , (−n, n). Whether will be arithmetic average η = ξi /n i=1 of random variables ξ1 , ξ2 , . . , ξn to converge in probability to zero under increasing of n? (5) At the spaceship the geiger for the definition of a number of hitting of the cosmic particles with the spaceship for some interval of time T is installed.
Q pq p2 q . . pm−1 q . . Since mξ = 1/p, then M [T ] = 2/p. 4 Exercises (1) At seismic stations A and B the seismograms are registered. Let x1 is the event consisting in a signal extraction with the probability p1 at a given time interval on a seismogram A, and x2 is the event of a signal extraction with the probability p2 at the station B. e. probability of a signal extraction or at the station A, or at the station B. (2) A subunit of a seismic station operates trouble-free during a random time T , the subunit renews after a failure.