=+20. In the family planning model of Example 6.6.2, let Msd be the number of children born when the family first attains either its quota of s sons or d daughters. Show that E(Msd) = E(min{Ts, Td}) = s−1 k=0



d−1 l=0

k + l k



pkql

.

Note that the formulas for E(max{Ts, Td}) and E(min{Ts, Td}) together yield E(min{Ts, Td}) + E(max{Ts, Td}) = E(Ts) + E(Td). (6.22)

Prove the general identity E(min{X, Y }) + E(max{X, Y }) = E(X) + E(Y )

for any pair of random variables X and Y with finite expectations.

Finally, argue that E(Msd) = d

s−1 k=0

d + k k



pkqd + s d

−1 l=0

s + l l



psql (6.23)

148 6. Poisson Processes by counting all possible successful sequences of births that lead to either the daughter quota or the son quota being fulfilled first. Combining equations (6.22) and (6.23) permits us to write E(Nsd) = s p

+

d q − d

s−1 k=0

d + k k



pkqd − s d

−1 l=0

s + l l



psql

, replacing a double sum with two single sums.