Consider the random variable YY
formed as the sum of two independent random variables
X
1
X
1
and
X
2
X
2
:
where
X
1
X
1
has pdf
f1
x1
f1
x1
and
X
2
X
2
has pdf
f2
x2
f2
x2
.
We can write the joint pdf for yy
and
x
1
x
1
by rewriting the conditional probability formula:
fy
x1
=fy|
x1
f1
x1
f
y
x1
f
|
y
x1
f1
x1
(2)
It is clear that the event '
YY
takes the value
yy conditional upon
X1
=
x1
X1
x1
' is equivalent to
X
2
X
2
taking a value
y−
x1
y
x1
(since
X2
=Y−
X1
X2
Y
X1
). Hence
fy|
x1
=
f2
y−
x1
f
|
y
x1
f2
y
x1
(3)
Now
fy
f
y
may be obtained using the
Marginal
Probability formula (
this equation from this discussion of
probability density
functions). Hence
fy=∫fy|
x1
f1
x1
d
x1
=∫
f2
y−
x1
f1
x1
d
x1
=
f2
*
f1
f
y
x1
f
|
y
x1
f1
x1
x1
f2
y
x1
f1
x1
f2
f1
(4)
This result may be extended to sums of three or more random
variables by repeated application of the above arguments for
each new variable in turn. Since convolution is a commutative
operation, for
nn independent
variables we get:
fy=
fn
*
f
n
-
1
*…*
f2
*
f1
=
fn
*
f
n
-
1
*…*
f2
*
f1
f
y
fn
f
n
-
1
…
f2
f1
fn
f
n
-
1
…
f2
f1
(5)
An example of this effect occurs when multiple dice are thrown
and the scores are added together. In the 2-dice example of the
subfigures a,b,c of
this figure in the discussion of probability
distributions, we saw how the pmf approximated a triangular
shape. This is just the convolution of two uniform 6-point pmfs
for each of the two dice.
Similarly if two variables with Gaussian pdfs are added
together, we shall show in the discussion of the summation of two or
more Gaussian random variables that this produces
another Gaussian pdf whose variance is the sum of the two input
variances.
