Page 46 - Elements of Distribution Theory

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052184472Xc01 CUNY148/Severini May 24, 2005 17:52

32 Properties of Probability Distributions

The function g(x) = I {x≤z} is bounded, but it is not continuous; however, it can be approx-
imated by a bounded, continuous function to arbitrary accuracy.
First suppose that X and Y are real-valued random variables. Fix a real number z and,
for > 0, define
1 if t ≤ z

g (t) ≡ g (t; z) = 1 − (t − z)/ if z < t < z + ;
0 if t ≥ z +
clearly g is bounded and continuous and
z+
E[g (X)] = F X (z) + [1 − (x − z)/ ] dF X (x)
z
where F X denotes the distribution function of X. Using integration-by-parts,

1 z+
E[g (X)] = F X (x) dx.
z
Hence, for all > 0,

1 z+ 1 z+
F X (x) dx = F Y (y) dy
z z
or, equivalently,

1 z+ 1 z+
F X (z) − F Y (z) = [F X (x) − F X (z)] dx − [F Y (y) − F Y (z)] dy.
z z
Since F X and F Y are non-decreasing,
1 z+ 1 z+
|F X (z) − F Y (z)|= [F X (x) − F X (z)] dx + [F Y (y) − F Y (z)] dy,
z z
and, hence, for all > 0,
|F X (z) − F Y (z)|≤ [F X (z + ) − F X (z)] + [F Y (z + ) − F Y (z)].

Since F X and F Y are right-continuous, it follows that F X (z) = F Y (z); since z is arbitrary, it
follows that F X = F Y and, hence, that X and Y have the same distribution.
The proof for the case in which X and Y are vectors is very similar. Suppose X and Y
d
d
take values in a subset of R .For a given value of z ∈ R , let
A = (−∞, z 1 ] ×· · · × (−∞, z p ]
and ρ(t)be the Euclidean distance from t to A.For > 0, define


 1 if ρ(t) = 0
g (t) ≡ g (t; z) = 1 − ρ(t)/ if 0 <ρ(t) < ;
0 if ρ(t) ≥

p
clearly g is bounded; since ρ is a continuous function on R , g is continuous as well.
Let X 0 = ρ(X) and Y 0 = ρ(Y); then X 0 and Y 0 are real-valued random variables, with
, respectively. Note that
distribution functions F X 0 and F Y 0

E[g (X)] = F X (z) + [1 − x/ ] dF X 0 (x)
0