Graphics Reference
In-Depth Information
these uses are consistent. Consider the identity function
I
:
S
→
S
:
s
→
s
.By
Equation 30.12, we have
p
I
(
t
)=
Pr
{
I
=
t
}
for
t
∈
S
.
(30.13)
For a fixed element
t
∈
S
, what is the event
I
=
t
? It's the event
{
s
∈
S
:
I
(
s
)=
t
}
=
{
s
∈
S
:
s
=
t
}
=
{
t
}
.
(30.14)
The probability of this event is just the sum of the probability masses for all out-
comes in the event, that is,
p
(
t
)
. Thus, the probability mass function for the identity
map is the same thing as the probability mass function
p
for the probability space
itself.
Functions like
Y
that send a probability space to some set
T
rather than
R
are
sometimes still called random variables. In the case we're most interested in,
Y
will send the unit square to the unit sphere or upper hemisphere, and we'll use
the terminology “random point.” We'll use the letter
Y
to denote random-point
functions, just as we use
X
for all the random variables in the chapter.
We conclude with one last bit of terminology. The function
p
Y
, whether
Y
is a
random variable or a map from
S
to some arbitrary set
T
, is called the
distribution
of
Y
; we say that
Y
is distributed according to
p
Y
. This suggestive terminology
is useful when we're thinking of things like student scores on a test, where it's
common to say, “The scores were distributed around a mean of 82.” In this case,
the underlying probability space
S
is the set of students, with a uniform probability
distribution on
S
, and the test score is a random variable from students to
R
. You'll
often see the notation
X
∼
f
to mean “X is a random variable with distribution
f
,”
that is, with
p
X
=
f
.
The expected value of a random variable
X
is a constant; it's often denoted
X
.
Note that
X
can also be regarded as a constant
function
on
S
, and hence be treated
as a random variable as well.
Expectation has two properties that we'll use frequently:
E
[
X
+
Y
]=
E
[
X
]+
E
[
Y
]
, and
E
[
cX
]=
c
E
[
x
]
for any real number
c
. In short,
expectation is linear
.
Expectation and variance (the latter which we'll encounter in a moment) are
higher-order functions: Their arguments are not numbers or points, but actual
functions—in particular, random variables.
When we write
E
[
X
+
Y
]
, we mean the expectation of the sum of the two
random variables
X
and
Y
, that is, the function
s
→
X
(
s
)+
Y
(
s
)
. This means
that when we write
X
+
X
, we mean the function
s
X
(
s
)+
X
(
s
)
.If
X
happens
to correspond to a piece of code that looks like
return randb(...)
, then you
cannot
implement
X
+
X
by
X() + X()
: The two invocations of the random
number generator may produce different values!
→
In general, when you look at a program with
randb
in it, your analysis of the
program will involve a probability space with 2
k
elements, where
k
is the number
of times
randb
is invoked; each element will have equal probability mass. The
same will be true when we look at
rand
: Any two numbers between 0 and 1 will
