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De Transcription | Bibliothèque patrimoniale numérique Mines ParisTech
the mean value of Z over v is the random variable Z dx . - x v v Clearly, we have: E(Z ) = — E ( f Zv dx"\ = — f E( Z )dx = V | v | \ v X / | v | v X = m Var(Zv) = E(Z2) - m2 ' * (U> '* "" ") " " = — f f EfZ, z„) dx dy - m2 IvI 2 'Vv x y JJ | V | 2 JvJv C(x-y) dx dy Now, let V be another quadrat of (R^ that contains v, and let Zv dénoté the mean value of Z over V
- v = — f zx dx | V | Jv
Then E(ZV) = m = E(Zv) which proves the assertion i) quoted in the introduction. On the other hand, Var(Z ) = —— f f C(x-y) dx dy | V| 2 J v* V Due to the averaging effects, one could expect that Var(Zv) ^ Var(Zv). However, this is not always the case. As a counter example, consider the monodimensional random function defined by:
