3.17 Let V be an open bounded region in d. If {X(t)} is a stationary random field...
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3.17 Let V be an open bounded region in ℝd. If {X(t)} is a stationary random field with covariance function ????(h), define the (continuous) sample mean within V by X(V) = |V|
−1
∫V X(t) dt.
Show that its variance is given by var{X(V)} = |V|
−2
∫V ∫V
????(s − t) ds dt.
By expanding the square for the sample continuous variance within V in the formula for the dispersion variance
????(0|V) = |V|
−1 E
{
∫ [X(t) − X(V)]2 dt}
, show that
????(0|V) = ????(0) − var{X(V)}.
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