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continuous rather than discrete so that the tools of calculus (e.g., instantaneous
rates) actually apply to it, while in fact only finite-time rate measurements (“19
cars arrived between 9:20 and 9:43”) make sense.
We'll do the same thing with light. As we observe the light arriving at a small
piece of surface, we can think of ourselves as counting “arriving photons over
some period of time.” But instead we treat the light as if it were infinitely divisible,
and talk about instantaneous rates of light arrival. In fact, rather than counting
photons, we'll count the arriving energy, because photons of different wavelengths
have different energies, but the idea remains the same.
This assumption that there is an instantaneous rate of energy arrival at a surface
lets us use calculus to talk about light energy. We'll repeat this “limiting trick”
twice more, once to establish a rate of arrival per area as we consider smaller and
smaller areas, and again to consider the rate of energy arriving from a particular
set of directions, divided by the size of that set of directions, as the size of the
set goes to zero. Having described this quantity (which we'll call radiance), we'll
see that all practical measurements we can make can be expressed as integrals
of radiance over various areas, time periods, and sets of directions. The abstract
entity, radiance, turns out to be easy to work with using calculus, and all the things
we can measure are integrals of radiance.
In this discussion so far, we've moved from a discrete version of counting to
one in which the light-energy arrival rate is continuous. We'll now do the same
thing in two more ways, with respect to angle and area.
26.6.1 A Brief Introduction to Probability Densities
Before we do so, let's look at a related concept from probability theory, the
notion of probability density. Consider a random number generator that ran-
domly generates real numbers between 0 and 5. We observe 1,000 of these ran-
domly generated real numbers, and look at how many lie between 0 and 0.5,
between 0.5 and 1.0, etc. The resultant histogram (see Figure 26.17) looks fairly
smooth; looking at it we might conjecture that the random number generator
is uniform, in the sense that every number is equally likely to be generated.
But if we choose smaller bins to count—say, between 0 and 0.001, between
0.0001 and 0.0002, etc.—the uniformity is no longer so obvious. Indeed, the
probability of generating any particular random number must be zero. Thus,
when we are discussing probabilities where the domain is some interval in the
reals (rather than a discrete set, like the set of faces on a pair of dice), we
talk not of the probabilities of generating particular numbers, but of generat-
ing numbers within an interval [ a , b ] . If the generator really is generating num-
bers uniformly, then the probability of generating a number in the interval [ a , b ]
is proportional to b
a . More generally, we posit the existence of a function
p :[ 0, 5 ]
R called the probability density function or pdf with the property
that
= b
a
p ( x ) dx .
Pr
{
a random number in the interval [ a , b ] is generated
}
For the uniform distribution on the interval [ 0, 5 ] , p is the constant function
with value 1
5. For other distributions, p is not constant. But because its integral
represents a probability, p must be everywhere nonnegative, and its integral over
[ 0, 5 ] must be 1. 0.
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