Rice J A Mathematical statistics and data analysis

frequency function just given can be reexpressed
as
pXY (x, y) = pX |Y (x |y)pY (y)
(the multiplication law of Chapter 1). This useful equation gives a relationship between
the joint and conditional frequency functions. Summing both sides over all values of
y, we have an extremely useful application of the law of total probability:
pX (x) =
∑
y
pX |Y (x |y)pY (y)
88 Chapter 3 Joint Distributions
E X A M P L E B Suppose that a particle counter is imperfect and independently detects each incoming
particle with probability p. If the distribution of the number of incoming particles in
a unit of time is a Poisson distribution with parameter λ, what is the distribution of
the number of counted particles?
Let N denote the true number of particles and X the counted number. From the
statement of the problem, the conditional distribution of X given N = n is binomial,
with n trials and probability p of success. By the law of total probability,
P(X = k) =
∞∑
n=0
P(N = n)P(X = k|N = n)
=
∞∑
n=k
λne−λ
n!
(
n
k
)
pk(1 − p)n−k
= (λp)
k
k!
e−λ
∞∑
n=k
λn−k
(1 − p)n−k
(n − k)!
= (λp)
k
k!
e−λ
∞∑
j=0
λ j (1 − p) j
j!
= (λp)
k
k!
e−λeλ(1−p)
= (λp)
k
k!
e−λp
We see that the distribution of X is a Poisson distribution with parameter λp. This
model arises in other applications as well. For example, N might denote the number
of traffic accidents in a given time period, with each accident being fatal or nonfatal;
X would then be the number of fatal accidents. ■
3.5.2 The Continuous Case
In analogy with the definition in the preceding section, if X and Y are jointly contin-
uous random variables, the conditional density of Y given X is defined to be
fY |X (y|x) = fXY (x, y)fX (x)
if 0 < fX (x) < ∞, and 0 otherwise. This definition is in accord with the result to
which a differential argument would lead. We would define fY |X (y|x) dy as P(y ≤
Y ≤ y + dy|x ≤ X ≤ x + dx) and calculate
P(y ≤ Y ≤ y + dy|x ≤ X ≤ x + dx) = fXY (x, y) dx dyfX (x) dx =
fXY (x, y)
fX (x) dy
Note that the rightmost expression is interpreted as a function of y, x being fixed.
The numerator is the joint density fXY (x, y), viewed as a function of y for fixed x :
you can visualize it as the curve formed by slicing through the joint density function
3.5 Conditional Distributions 89
perpendicular to the x axis. The denominator normalizes that curve to have unit
area.
The joint density can be expressed in terms of the marginal and conditional
densities as follows:
fXY (x, y) = fY |X (y|x) fX (x)
Integrating both sides over x allows the marginal density of Y to be expressed as
fY (y) =
∫ ∞
−∞
fY |X (y|x) fX (x) dx
which is the law of total probability for the continuous case.
E X A M P L E A In Example D in Section 3.3, we saw that
fXY (x, y) = λ2e−λy, 0 ≤ x ≤ y
fX (x) = λe−λx , x ≥ 0
fY (y) = λ2 ye−λy, y ≥ 0
Let us find the conditional densities. Before doing the formal calculations, it is in-
formative to examine the joint density for x and y, respectively, held constant. If x
is constant, the joint density decays exponentially in y for y ≥ x ; if y is constant,
the joint density is constant for 0 ≤ x ≤ y. (See Figure 3.7.) Now let us find the
conditional densities according to the preceding definition. First,
fY |X (y|x) = λ
2e−λy
λe−λx
= λe−λ(y−x), y ≥ x
The conditional density of Y given X = x is exponential on the interval [x,∞).
Expressing the joint density as
fXY (x, y) = fY |X (y|x) fX (x)
we see that we could generate X and Y according to fXY in the following way: First,
generate X as an exponential random variable ( fX ), and then generate Y as another
exponential random variable ( fY |X ) on the interval [x,∞). From this representation,
we see that Y may be interpreted as the sum of two independent exponential random
variables and that the distribution of this sum is gamma, a fact that we will derive
later by a different method.
Now,
fX |Y (x |y) = λ
2e−λy
λ2 ye−λy
= 1
y
, 0 ≤ x ≤ y
The conditional density of X given Y = y is uniform on the interval [0, y]. Finally,
expressing the joint density as
fXY (x, y) = fX |Y (x |y) fY (y)
90 Chapter 3 Joint Distributions
we see that alternatively we could generate X and Y according to the density fXY by
first generating Y from a gamma density and then generating X uniformly on [0, y].
Another interpretation of this result is that, conditional on the sum of two independent
exponential random variables, the first is uniformly distributed. ■
E X A M P L E B Stereology
In metallography and other applications of quantitative microscopy, aspects of a three-
dimensional structure are deduced from studying two-dimensional cross sections.
Concepts of probability and statistics play an important role (DeHoff and Rhines
1968). In particular, the following problem arises. Spherical particles are dispersed
in a medium (grains in a metal, for example); the density function of the radii of
the spheres can be denoted as fR(r). When the medium is sliced, two-dimensional,
circular cross sections of the spheres are observed; let the density function of the radii
of these circles be denoted by fX (x). How are these density functions related?
x
y
H
x
r
F I GUR E 3.13 A plane slices a sphere of radius r at a distance H from its center,
producing a circle of radius x .
To derive the relationship, we assume that the cross-sectioning plane is chosen
at random, fix R = r , and find the conditional density fX |R(x |r). As shown in
Figure 3.13, let H denote the distance from the center of the sphere to the planar cross
section. By our assumption, H is uniformly distributed on [0, r ], and X = √r 2 − H 2.
We can thus find the conditional distribution of X given R = r :
FX |R(x |r) = P(X ≤ x)
= P(
√
r 2 − H 2 ≤ x)
= P(H ≥
√
r 2 − x2)
= 1 −
√
r 2 − x2
r
, 0 ≤ x ≤ r
3.5 Conditional Distributions 91
Differentiating, we find
fX |R(x |r) = x
r
√
r 2 − x2 , 0 ≤ x ≤ r
The marginal density of X is, from the law of total probability,
fX (x) =
∫ ∞
−∞
fX |R(x |r) fR(r) dr
=
∫ ∞
x
x
r
√
r 2 − x2 fR(r) dr
[The limits of integration are x and ∞ since for r ≤ x , fX |R(x |r) = 0.] This equation
is called Abel’s equation. In practice, the marginal density fX can be approximated
by making measurements of the radii of cross-sectional circles. Then the problem
becomes that of trying to solve for an approximation to fR , since it is the distribution
of spherical radii that is of real interest. ■
E X A M P L E C Bivariate Normal Density
The conditional density of Y given X is the ratio of the bivariate normal density to a
univariate normal density. After some messy algebra, this ratio simplifies to
fY |X (y|x) = 1
σY
√
2π(1 − ρ2) exp
⎛⎜⎜⎜⎝−12
[
y − μY − ρ σY
σX
(x − μX )
]2
σ 2Y (1 − ρ2)
⎞⎟⎟⎟⎠
This is a normal density with mean μY + ρ(x −μX )σY /σX and variance σ 2Y (1 − ρ2).
The conditional distribution of Y given X is a univariate normal distribution.
In Example B in Section 2.2.3, the distribution of the velocity of a turbulent
wind flow was shown to be approximately normally distributed. Van Atta and Chen
(1968) also measured the joint distribution of the velocity at a point at two different
times, t and t + τ . Figure 3.14 shows the measured conditional density of the ve-
locity, v2, at time t + τ , given various values of v1. There is a systematic departure
from the normal distribution. Therefore, it appears that, even though the velocity
is normally distributed, the joint distribution of v1 and v2 is not bivariate normal.
This should not be totally unexpected, since the relation of v1 and v2 must con-
form to equations of motion and continuity, which may not permit a joint normal
distribution. ■
Example C illustrates that even when two random variables are marginally