TELEPHONE TRAFFIC TIME AVERAGES 1139 



The fourth moment is given by 



E[M\T)] = ^ / dt du dv dw (pit — u, u — V, V — w) 

 1^ Jo Jo Jo Jq 



= b' 



"t" rp4 



24 ' ''' "' "" ^" 



b^ / / dtdudv dw\g{t - w) + g{l - v) 



Jo Jo Jo Jo 



+ git' — w) + giu — v) -{■ giu — w) + giv — w)] 



T pt /.M /.t? 



f J pi pU pV 



+ &M / / I dtdudv dw [git -v)+giu-w)-[- 2git - w)] 

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-\- b I I I dtdudv dw [git — u) giv —w) -{- 



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git - v)giu - w) -\- git - w)giu - v)] 



+ b I I I dtdudvdw[git - w)] 



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Employing the identities 



• T pT pT pT 



4 / / / dtdudv dw [git - u) + git - v) + git - w) 



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+ giu — v) -}- giu - w) + giv — w)] 



pT pt pu pv 



= / / / I dt du dv dw gi\t — u\) 

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= 2T' i dxgix)iT - x) = fh/b, 

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pT pt pu pv 



8 / / / I dt du dv dw [git — u) giv — w) -{- git — v) giu — w) 

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+ git - w) giu — v)] 



= I I / / dt dw dv dw gi\t — u \)gi\ v — w\) 

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= 4 [£ dx g{x)(T - x)^ = tk\lh\ 



and successive inversion of order of integration, the final result turns out 

 to be 



