....and well, regarding the following, dazzlenation....no matter how fascinating the tale....i'm looking at the 'wallpaper' here....lol....because things are often not as they appear, eh - worf?....see previous posts....go figure:
Some authors require each value {\displaystyle w(t)}w(t) to be a real-valued random variable with expectation {\displaystyle \mu }\mu and some finite variance {\displaystyle \sigma ^{2}}\sigma ^{2}. Then the covariance {\displaystyle \mathrm {E} (w(t_{1})\cdot w(t_{2}))}\mathrm {E} (w(t_{1})\cdot w(t_{2})) between the values at two times {\displaystyle t_{1}}t_{1} and {\displaystyle t_{2}}t_{2} is well-defined: it is zero if the times are distinct, and {\displaystyle \sigma ^{2}}\sigma ^{2} if they are equal. However, by this definition, the integral
{\displaystyle W_{[a,a+r]}=\int _{a}^{a+r}w(t)\,dt}W_{[a,a+r]}=\int _{a}^{a+r}w(t)\,dt
over any interval with positive width {\displaystyle r}r would be simply the width times the expectation: {\displaystyle r\mu }{\displaystyle r\mu }. This property would render the concept inadequate as a model of physical "white noise" signals.
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