Mathematical Methods for Physics and Engineering - Matematica.NET

PROBABILITYScaling and shiftingIf Y = aX + b, wherea and b are arbitrary constants, thenM Y (t) =E [ e tY ] = E [ e t(aX+b)] = e bt E [ e atX] = e bt M X (at). (30.88)This result is often useful for obtaining the central moments of a distribution. If theMFG of X is M X (t) then the variable Y = X−µ has the MGF M Y (t) =e −µt M X (t),which clearly generates the central moments of X, i.e.( )E[(X − µ) n ]=E[Y n ]=M (n) dnY (0) = dt n [e−µt M X (t)] .t=0Sums of random variablesIf X 1 ,X 2 ,...,X N are independent random variables and S N = X 1 + X 2 + ···+ X NthenM SN (t) =E [ e ] tSN = E [ [ N]e t(X1+X2+···+XN)] ∏= E e tXi .Since the X i are independent,N∏M SN (t) = E [ e tXi] N∏= M Xi (t). (30.89)i=1In words, the MGF of the sum of N independent random variables is the productof their individual MGFs. By combining (30.89) with (30.88), we obtain the moregeneral result that the MGF of S N = c 1 X 1 + c 2 X 2 + ···+ c N X N (where the c i areconstants) is given byM SN (t) =i=1i=1N∏M Xi (c i t). (30.90)i=1Variable-length sums of random variablesLet us consider the sum of N independent random variables X i (i =1, 2,...,N), allwith the same probability distribution, and let us suppose that N is itself a randomvariable with a known distribution. Following the notation of section 30.7.1,S N = X 1 + X 2 + ···+ X N ,where N is a random variable with Pr(N = n) =h n and probability generatingfunction χ N (t) = ∑ h n t n . For definiteness, let us assume that the X i are continuousRVs (an analogous discussion can be given in the discrete case). Thus, the1164