Sometimes
you need to know the distribution of some combination of things. The sum of two
incomes, for example, or the difference between demand and capacity. If f_{X}(x)
is the distribution (probability density function, pdf) of one item, and f_{Y}(y)
is the distribution of another, what is the distribution of their sum, Z = X + Y ?
As a simple example consider X and Y to have a uniform
distribution on the interval (0, 1). The distribution of their sum is triangular on
(0, 2).
Why? To begin consider the problem qualitatively. The minimum possible
value of Z = X + Y is zero when x=0 and y=0, and the maximum possible value is two, when
x=1 and y=1. Thus the sum is defined only on the interval (0, 2) since the probability of
z<0 or z>2 is zero, that is,
P(Z  z<0) = 0 and P(Z  z>2) = 0.
Further, it seems
intuitive^{(1)} that the most
probable value would be near z=1, the midpoint of the interval, for several reasons. The
summands are iid (independent, identically distributed) and the sum is a
linear operation that doesn't distort symmetry. So we would intuit^{(2)} that the probability density of
Z = X + Y should start at zero at z=0, rise to a maximum at midinterval, z=1, and then
drop symmetrically to zero at the end of the interval, z=2. We might expect the
distribution of
Z = X + Y to look like this:
Enough of visceral pseudo calculus. How do
you prove that this result is correct?
Proof:
F_{Z}(z), the cdf of Z, is the probability that the sum, Z, is less than or equal
to some value z. The probability density that we're looking for is
f_{Z}(z) = d[F_{Z}(z)]/dz, by relationship between a cdf and a pdf.
