Mathematical Methods for Physics and Engineering - Matematica.NET

PROBABILITYnr = nrFigure 30.10The pairs of values of n and r used in the evaluation of Φ X+Y (t).Sums of random variablesWe now turn to considering the sum of two or more independent randomvariables, say X and Y , and denote by S 2 the random variableS 2 = X + Y.If Φ S2 (t) isthePGFforS 2 , the coefficient of t n in its expansion is given by theprobability that X + Y = n and is thus equal to the sum of the probabilities thatX = r and Y = n − r for all values of r in 0 ≤ r ≤ n. Since such outcomes fordifferent values of r are mutually exclusive, we havePr(X + Y = n) =∞∑Pr(X = r)Pr(Y = n − r). (30.79)r=0Multiplying both sides of (30.79) by t n and summing over all values of n enablesus to express this relationship in terms of probability generating functions asfollows:∞∑∞∑ n∑Φ X+Y (t) = Pr(X + Y = n)t n = Pr(X = r)t r Pr(Y = n − r)t n−rn=0=n=0 r=0∞∑r=0 n=r∞∑Pr(X = r)t r Pr(Y = n − r)t n−r .The change in summation order is justified by reference to figure 30.10, whichillustrates that the summations are over exactly the same pairs of values of n andr, but with the first (inner) summation over the points in a column rather thanover the points in a row. Now, setting n = r + s gives the final result,Φ X+Y (t) =∞∑Pr(X = r)t rr=0∞ ∑s=0Pr(Y = s)t s=Φ X (t)Φ Y (t), (30.80)1160