Fall 2002 - Course 3 Solutions

Question #15
Answer: C
Let:
N = number
X = profit
S = aggregate profit
subscripts G = “good”, B = “bad”, AB = “accepted bad”
c hb g
1 1
c 2hc 3hb60g 10 (If you have trouble accepting this, think instead of a heads-tails rule, that
2
λ G =
3
60 = 40
λ AB = =
the application is accepted if the applicant’s government-issued identification number, e.g. U.S.
Social Security Number, is odd. It is not the same as saying he automatically alternates
accepting and rejecting.)
2
b Gg b Gg b Gg b Gg b Gg
= b40gb10, 000g+ b40gd300 i
2 = 4, 000,
000
2
b ABg b ABg b ABg b ABg b ABg
Var S = E N × Var X + Var N × E X
Var S = E N × Var X + Var N × E X
b gb g b gb g 2
= 10 90, 000 + 10 − 100 = 1, 000,
000
S G and S AB are independent, so
b g b g b g
Var S = Var SG
+ Var S AB = 4, 000, 000 + 1, 000,
000
= 5, 000,
000
If you don’t treat it as three streams (“goods”, “accepted bads”, “rejected bads”), you can
compute the mean and variance of the profit per “bad” received.
1
λ B =
3
60 = 20
c hb g
d i b g b g
2
b g
2 B 2
B B
If all “bads” were accepted, we would have E X = Var X + E X
= 90, 000+ − 100 = 100,
000
Since the probability a “bad” will be accepted is only 50%,
E X = Prob accepted · E X accepted + Prob not accepted · E X not accepted
b Bg b g c B h b g c B h
b gb g b gb g
d i b gb g b gb g
E X B
2
Likewise,
= 05 . − 100 + 05 . 0 = −50
= 05 . 100, 000 + 05 . 0 = 50,
000
2
b Bg b Bg b Bg b Bg b Bg
2
= b20gb47, 500g + b20gd50 i = 1000 , , 000
Now Var S = E N × Var X + Var N × E X
S G and S B are independent, so
b g b g b g
Var S = Var SG
+ Var SB
= 4, 000, 000 + 1, 000,
000
= 5, 000,
000