Let X have the moment generating function of Example 3.29 and let Y = X − 1. Recall that X is the number of people who need to be checked to get someone who is Rh+, so Y is the number of people checked before the first Rh+ person is found. Find MY(t) using the last proposition in this section.

Example 3.29

Example 3.10

Example 3.29 Consider testing individuals' blood samples one by one in order to find someone whose blood type is Rh+. The rv X = the number of tested samples should follow the pmf specified in Example 3.10 with p = .85: p(x) = .85(.15)-1 for x = 1,2,3,.... Determining the moment generating function here requires using the formula for the sum of a geometric series: 1+r++...= = 1/(1-r) for r] < 1. The moment generating function is Mx (t) = E(ex) = exp(x) = e.85(.15)*- = .85ee(x-1)(.15)*-1 XED x=1 x=1 = .85e(.15e)- = .85e[1 +.15e + (.15e) + -] - x=1 .85et 1.15et The condition on r requires .15e| < 1. Dividing by .15 and taking logs gives t