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 fX(x)
is the distribution (probability density function, pdf) of one item, and fY(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
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 mid-interval, z=1, and then
drop symmetrically to zero at the end of the interval, z=2. We might expect the
Z = X + Y to look like this:
Enough of visceral pseudo calculus. How do
you prove that this result is correct?
FZ(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
fZ(z) = d[FZ(z)]/dz, by relationship between a cdf and a pdf.