5Continuous random variables
IA Probability
5.2 Stochastic ordering and inspection paradox
We want to define a (partial) order on two different random variables. For
example, we might want to say that
X
+ 2
> X
, where
X
is a random variable.
A simple definition might be expectation ordering, where
X ≥ Y
if
E
[
X
]
≥
E
[
Y
]. This, however, is not satisfactory since
X ≥ Y
and
Y ≥ X
does not imply
X = Y . Instead, we can use the stochastic order.
Definition (Stochastic order). The stochastic order is defined as:
X ≥
st
Y
if
P(X > t) ≥ P(Y > t) for all t.
This is clearly transitive. Stochastic ordering implies expectation ordering,
since
X ≥
st
Y ⇒
Z
∞
0
P(X > x) dx ≥
Z
∞
0
P(y > x) dx ⇒ E[X] ≥ E[Y ].
Alternatively, we can also order things by hazard rate.
Example (Inspection paradox). Suppose that
n
families have children attending
a school. Family
i
has
X
i
children at the school, where
X
1
, ··· , X
n
are iid
random variables, with
P
(
X
i
=
k
) =
p
k
. Suppose that the average family size is
µ.
Now pick a child at random. What is the probability distribution of his
family size? Let
J
be the index of the family from which she comes (which is a
random variable). Then
P(X
J
= k | J = j) =
P(J = j, X
j
= k)
P(J = j)
.
The denominator is 1
/n
. The numerator is more complex. This would require
the
j
th family to have
k
members, which happens with probability
p
k
; and
that we picked a member from the
j
th family, which happens with probability
E
h
k
k+
P
i=j
X
i
i
. So
P(X
J
= k | J = j) = E
"
nkp
k
k +
P
i=j
X
i
#
.
Note that this is independent of j. So
P(X
J
= k) = E
"
nkp
k
k +
P
i=j
X
i
#
.
Also, P(X
1
= k) = p
k
. So
P(X
J
= k)
P(X
1
= k)
= E
"
nk
k +
P
i=j
X
i
#
.
This is increasing in
k
, and greater than 1 for
k > µ
. So the average value of the
family size of the child we picked is greater than the average family size. It can
be shown that X
J
is stochastically greater than X
1
.
This means that if we pick the children randomly, the sample mean of the
family size will be greater than the actual mean. This is since for the larger a
family is, the more likely it is for us to pick a child from the family.