Theory of Statistics - George Mason University

1.1 Some Important Probability Facts 57
Id Id
A = ,
0 Id
so that we have a full-rank transformation, (U, V ) T = A(X, Y ) T The inverse
of the transformation matrix is
A −1
Id −Id
= ,
0 Id
and the Jacobian is 1. Because X and Y are independent, their joint PDF
is fXY (x, y) = fX(x)fY (y), and the joint PDF of U and V is fUV (u, v) =
fX(u − v)fY (v); hence, the PDF of U is
fU(u) =
IR d fX(u − v)fY (v)dv
=
IR d fY (u − v)fX(v)dv.
(1.124)
We call fU the convolution of fX and fY . The commutative operation of
convolution occurs often in applied mathematics, and we denote it by fU =
fX ⋆ fY . We often denote the convolution of a function f with itself by f (2) ;
hence, the PDF of X1 + X2 where X1, X2 are iid with PDF fX is f (2)
X . From
equation (1.124), we see that the CDF of U is the convolution of the CDF of
one of the summands with the PDF of the other:
FU = FX ⋆ fY = FY ⋆ fX. (1.125)
In the literature, this operation is often referred to as the convolution of the
two CDFs, and instead of as in equation (1.125), may be written as
FU = FX ⋆ FY .
Note the inconsistency in notation. The symbol “⋆” is overloaded. Following
the latter notation, we also denote the convolution of the CDF F with itself
as F (2) .
Example 1.14 sum of two independent Poissons
Suppose X1 is distributed as Poisson(θ1) and X2 is distributed independently
as Poisson(θ2). By equation (1.124), we have the probability function of the
sum U = X1 + X2 to be
fU(u) =
u
v=0
θ u−v
1 eθ1 θ
(u − v)!
v 2eθ2 v!
= 1
u! eθ1+θ2 (θ1 + θ2) u .
The sum of the two independent Poissons is distributed as Poisson(θ1 +θ2).
Theory of Statistics c○2000–2013 James E. Gentle