kniga 7 - Probability and Statistics 1 - Sheynin, Oscar

( N −1)((S − N )(2NS− 3S− 3N+ 3) S(µ34 + 3µ 2 2 ),N ( S −1)(S − 2)( S − 3)U [N/S] 3( N −1)(N − 2)( N − 3) S22 =µ 2 32 +N ( S −1)(S − 2)( S − 3)( N −1)(S − N )( NS − S − N −1)S(µ34 + 3µ 2 2 ).N ( S −1)(S − 2)( S − 3)Introducing, as we did before, the notation (10), we can also represent these formulas inthe following way:U [N/S] [ −2]2N S2 = µ2 [ −2]2 , U [N/S] [ −3]3N S3 =3 [ −3]N SN Sµ 3 ,U [N/S] [ −4]4N S N4 = µ4 [ −4]4 +N S[ −2]U [N/S] [ −4]4N S22 = µ 2 N4 [ −4]2 +N S2S ( S − N )(2NS− 3S− 3N+ 3)(µ2 [ −2]24 + 3µ 2 2 ),N S N ( S − 2)( S − 3)[ −2]2S ( S − N )( NS − S − N −1)(µ2 [ −2]24 + 3µ 2 2 ).N S N ( S − 2)( S − 3)It is useful to note the following relation between the parameters {} and {µ}: 4 + 3 22 = [S/(S – 1)] (µ 4 + 3µ 2 2 ), (13)or, which is the same,S [– 2] ( 4 + 3 22 ) = S 2 (µ 4 + 3µ 2 2 ).7. Such is the general method that enables us to throw a bridge from the formulasconcerning the scheme of the returned ticket to those describing the other pattern. In somecases, however, the formulas sought can be also obtained in a shorter way. Thus, in theexamples above, the following reasoning will rapidly lead us to our goal. And thesupplementary calculational importance of the parameters {M} will reveal itself with specialclearness. We issue from the remark that for N = S and denotingx (S) = (1/S)(x 1 + x 2 + … + x S ) = m 1we haveU r [S/S] = E(1/S)U[ S / S ]r1r2 ... rmSi = 1= E{[(1/S)µ ...(x i – x S ) r = E(1/S)Si = 1rµrµ1 2 r m.( x − m ) 1ir1Si = 1(x i – m 1 ) r = Eµ r = µ r ,S] … [(1/S)i = 1(1x )r mi− m ]} =[ S / S ]r1r2 ... rmTaking this into account and, on the other hand, expressing U through theparameters {}, we shall have