366 Frequently Asked Questions In Quantitative Finance
the domain in the upper right ‘quadrant’ between the
point (0, 0,..., 0) and the planex 1 +x 2 +...+xn=1.
This is just
∫ 1
0
∫ 1 −x 1
0
∫ 1 −x 1 −x 2
0
...
∫ 1 −x 1 −x 2 −...−xn− 1
0
1 dxn...dx 3 dx 2 dx 1.
After doing several of the inner integrals you will find
that the answer is simplyn^1 !.
From this it follows that the probability that the sum
goes over one for the first time on thenth random
variable is
(
1 −
1
n!
)
−
(
1 −
1
(n−1)!
)
=
n− 1
n!
.
The required expectation is the sum ofn(n−1)/n!=
1 /(n−2)! from two to infinity, or equivalently the sum
of 1/n!fornzero to infinity. And this is our answer,e.
Minimum and maximum correlation
IfX,YandZare three random variables such that
XandYhave a correlation of 0.9, andYandZhave
correlation of 0.8, what are the minimum and maximum
correlation thatXandZcan have?
(Thanks to jiantao.)
Solution
The correlation matrix
1 ρXY ρXZ
ρXY 1 ρYZ
ρXZ ρYZ 1
must be positive semi definite. A bit of fooling around
with that concept will result in the following
