Relation Between Central and Non - Central Moments Let X be a RV discrete or continuous Let Un = E{(X - a)"}, be a non - central moment Un = (-1)" . "Cr ' My-r . d' 1=0 and Un = E{(X - H)" }, be a central moment, where u = E(X) Un = "Co . uh . do - "C1 . Un-1 . d + "C2 . Un-2 . d2 M1 = E(X - a) -"C3 ' Hin-3 . d3 + ... + (-1)" . "Cn . Ho . d" Mi = E(X) - E(a) Un = H'n -"C1 . Uh-1 . d + "C2 . un-2 . d2 H1 = u - a (: E(X) = u and E(constant) = constant) -"C3 ' Mini-3 . d3 + ... + (-1)". dn Mi = d (say) ... (1) where, Ho =1 and #j = d =u-a Now Un = E{(X - M)" } In particular, H1 = M1 - 161 . Ho . d = d - d =0 Un = E{(X - a - u + a)"} H2 = 12 - 201 . Hj . d + 2C2 . Ho . d2 =12 - 2d2 + d2 Un = E{([X - a] - [u - a])"} 12 = 12 - d2 Un = E{([X - a] - d)"} (from (1)) M3 = 13 - 3C1 . H2 . d + 362 . Mi . d2 - 303 . Ho . d3 Now we know that binomial expansion of (a - b)" = > (-1)" . "Cr . an-r . b" =M3 - 3 . H2 . d + 3d3 - d3 M3 = M3 -3u2d+ 2d3 Hn = E (-1) . Cy . ( x - a)-r . dar H4 = M'4 -4C1 . M's . d +462. 12 . d2 -+C3 . Hi . d3 + + C 4 . HO . d4 Un = (-1) . C, . E( (X - a) "- ] . d' =H'4- 4 . 13 . d + 6 . 12 . d2 - 4d4+ d4 un = E((X - a) " } = Un-r = E{ (X- a)"-r} H4 = M'4 - 4ugd+ 6/2d2 - 3d4