Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 902 — #910
902
Chapter 20
Deviation from the Mean
(c) Given this setup (and assuming mutual independence between the loans), what
is the expected failure rate in the mezzanine tranche?
(d) We take the mezzanine tranches from 100 bonds and create a CDO. What is
the expected number of underlying failures to hit the CDO?
(e) We divide this CDO into 10 tranches of 1000 bonds each. Assuming mutual
independence, give an upper bound on the probability of one or more failures in the
best tranche. The third tranche?
(f) Repeat the previous question without the assumption of mutual independence.
Homework Problems
Problem 20.34.
We have two coins: one is a fair coin, but the other produces heads with probability
3=4. One of the two coins is picked, and this coin is tossed n times. Use the
Chernoff Bound to determine the smallest n which allows determination of which
coin was picked with 95% confidence.
Problem 20.35.
An infinite version of Murphy’s Law is that if an infinite number of mutually independent
events are expected to happen, then the probability that only finitely many
happen is 0. This is known as the first Borel-Cantelli Lemma.
(a) Let A 0 ; A 1 ; : : : be any infinite sequence of mutually independent events such
that
X
PrŒA n D 1: (20.29)
Prove that PrŒno A n occurs D 0.
n2N
Hint: B k the event that no A n with n k occurs. So the event that no A n occurs is
B WWD \ k2N
B k :
Apply Murphy’s Law, Theorem 20.6.4, to B k .
(b) Conclude that PrŒonly finitely many A n ’s occur D 0.
Hint: Let C k be the event that no A n with n k occurs. So the event that only
finitely many A n ’s occur is
C WWD [ k2N
C k :